A Catalog of the Real Numbers - Carnegie Mellon...

A Catalog of the Real Numbers

by Michael Ian Shamos, Ph.D., J.D.

School of Computer Science Carnegie Mellon University

2007

Preface

Have you ever looked at the first few digits the decimal expansion of a number and tried to recognize which number it was? You probably do it all the time without realizing it. When you see 3.14159... or 2.7182818... you have no trouble spotting and e, respectively. But are

you on familiar terms with 0.572467033... or 0.405465108...? These are 2 3

12

and

, both of which have interesting stories to tell. You will find them, along with those of more than 10,000 other numbers, listed in this book. But why would someone bother to compile a listing of such facts, which may seem on the surface to be a compendium of numerical trivia?

log log3 2

Unless you had spent some time with 2 3

12

, you might not realize that it is the sum of two

very different series, ( )cos

12

1

k

k

k

k and . The proof that each sum

equals

1

)12()2(k

kkk

2 3

12

is elementary; what connection there may be between the two series remains

elusive. The number log turns up as the sum of several series and the value of certain

definite integrals:

log3 2

1

31k

k k

, ( )

1

2

1

1

k

kk k

,

)2)(1 xx

dx

1 ( and

0 2 xe

dx

1 . If your research led to

either number, this information might well lead you to additional findings or help you explain what you had already discovered.

Mathematics is an adventure, and a very personal one. Its practitioners are all exploring the same territory, but without a roadmap they all follow different paths. Unfortunately, the mathematical literature dwells primarily on final results, the destinations, if you will, rather than the paths the explorers took in arriving at them. We are presented with an orderly sequence of definitions, lemmas, theorems and corollaries, but are often left mystified as to how they were discovered in the first place.

The real empirical process of mathematics, that is, the laborious trek through false conjectures and blind alleys and taking wrong turns, is a messy business, ill-suited to the crisp format of referred journals. Possibly the unattractiveness of the topic accounts for the scarcity of writing on the subject. How many unsuccessful models of computation did Turing invent before he hit on the Turing machine? How did he even generate the models? How did he realize that the Halting Problem was interesting to consider, let alone imagine that it was unsolvable? What did his scrap paper look like?

This book is a field guide to the real numbers, similar in many ways to a naturalist’s handbook. When a birdwatcher spots what he thinks may be a rare species, he notes some of its characteristics, such as color or shape of beak, and looks it up in a field guide to verify his identification. Likewise, when you see part of a number (its initial decimal digits), you should be able to look it up here and find out more about it. The book is just a lexicographic list of 10,000 numbers arranged by initial digits of their decimal expansions, along with expressions whose values share the same initial digits.

PREFACE iv

This book was inspired by Neil Sloane’s A Handbook of Integer Sequences. That work, first published in 1971, is an elaborately-compiled list of the initial terms of over 2000 integer sequences arranged in lexicographic order. When one is confronted with the first few term of an unfamiliar sequence, such as {1, 2, 5, 16, 65, 326, 1957,...}, a glance at Sloane reveals that this sequence, number 589, is the total number of permutations of all subsets of n objects, that

is,

n

k

n

k

n

k jn

kn

n

k

nk

111 !

1!

)!(

!! , which is the closest integer to n . Sloane’s book is

extremely useful for finding relationships among combinatorial problems. It leads to discoveries, which help in turn generate conjectures that hopefully lead to theorems.

e!

What impelled me to compile this catalog was an investigation into Riemann’s -function that I began in 1990. It struck me as incredible that we have landed men on the moon but still

have not found a closed-form expression for ( )31

31

kk

. I say incredible because in the 18th

century Euler proved that ( )sk s

k

1

1

is a rational multiple of s for all even s, and gave a

formula for the coefficient in terms of Bernoulli numbers. No such formula is known for any odd s. It is not even known whether a closed form exists.

I began exploring such sums as 1

2 k ks sk t

, 1

1 k ks sk t

and ( )

1

2

k

s sk k k t

, whose values

can often be written as simple expressions involving various values of the -function. For

example, 1 5

43

5 32 k kk

( ), 1

6

1

24 21

2

k kk

coth and the remarkable

1 1

124 22 k kk

. To keep track of these many results, I kept them in a list by numerical

value, conveniently obtained using Mathematica®. Very quickly, after experimenting with

sums of the form , I began to notice connections that led to the following

theorem.

( )ak bk

11

Theorem 1. For an integer, b an integer such that a ba 1 2 and c a positive real

number, then 1

1 1ck kbj

aj b

ca bk

j

( )

For example, when I observed that the sums

.

An proof is given at the end of the chapter. The proof is elementary, but I never would have been motivated to write down the theorem in the first place without studying the emerging list of expressions sorted by numerical value.

After a time, I became somewhat obsessed with expanding the list of values and expressions and found that as it grew it became more and more useful. I began to add material not directly related to my research and found that it raised more questions than it answered. It became for me both a handbook and a research manifesto.

1

221

k

k

and ( )2

4 11

kk

k

of the form

shared the same 20-

digit prefix, this evoked considerable interest, since sums1

0 abk

k

are very difficult

PREFACE v

to treat analytically. After verifying that the sums w ual to ov

look for generalizations, exploring such sums as

ere eq er 500 digits, I began to 1

230

k

k ,

1

440

k

k ,

( )2

1

k

21k

k and

( )3

4 11

kk

k

give a few examples. The results suggested a conjecture, which stheorem:

, to

oon yielded the following

Theorem 2. For c > 1 real and p prime, 1

11 1c cpi

kp

kp

k

i

(, where

) ( )n is the

unction.

For a proof, see the end of the chapter. Alternatively, the theorem may be expressed in the

form

Moebius f

1

1

1

)(

kka

kp

1 /1kpp

k

a . This allows us to relate such strange expressions as

12

1k and

k 3

1k to Moebius function sums. Unfortunately, the theorem seems to

1 3k e

shed no light on trickier sums like 12

1 kk

k

and 1

1 k kk

k .

This b what is kna small space a potentially huge number of objects. It is similar to the drawers in the card catalog of a library. Imagine that a library only has 26 card drawers and that it places catalog

them. nd a b k by Taylor, you must go to drawer “T” and

ook is own in computer science as a hash table, a structure for indexing in

cards in each draw without sorting To fi oo look through all of the cards. If the library has only a hundred books and the authors’

names are reasonably distributed over the alphabet, you can expect to have to look at only two or three cards to find the book you want (if the library has it). If the library has a million books, this method would be impractical because you would have to leaf through more than 20,000 cards on the average. That’s because about 40,000 cards will “hash” to the same drawer. However, if the library had more card drawers, let’s say 17,576 of them labeled “AAA” through “ZZZ,” you would have a much easier time since each drawer would only have about 60 cards and you would expect to leaf through 30 of them to find a book the library holds (all 60 to verify that the book is not there).

Hashing was originally used to create small lookup tables for lengthy data elements. The term “hashing” in this context means cutting up into small pieces. Suppose that a company has 300 employees and used social security numbers (which have nine digits) to identify them. A table large enough to accommodate all possible social security numbers would need 109 entries, too large even to store on most computer hard disks. Such a table seems wasteful anyway, since only a few hundred locations are going to be occupied. One might try using just the first three digits of the number, which reduces the space required to just a thousand locations. A drawback is that the records for several employees would have to be stored at that location (if their social security numbers began with the same three digits). An attempt to store a record in a location where one already exists is called a “collision” and must be resolved using other methods. There will not be many collisions in a sparse table unless the numbers tend to cluster strongly (as the initial digits of social security numbers do). Using this scheme in a small community where workers have lived all their lives might result in everyone hashing

PREFACE vi

to the same location because of the geographical scheme used to assign social security numbers. A better method would be to use the three least-significant digits or to construct a “hash function,” one that is designed to spread the records evenly over the storage locations. A suitable hash function for social security numbers might be to square them and use the middle three digits. (I haven’t tried this.)

Getting back to the library, notice that any two cards in a drawer may be identical (for multiple copies of the same book) or different. The only thing they are guaranteed to have in common is the first three letters of the author’s last name. Now imagine that the library sets up 2620 drawers. Aside from the fact that you might have to travel for some time to get to a dra

authors have the same name, even if their nam

wer, once you arrive there you will probably find very few cards in it, and the chances are excellent that if there is more than one card, you are dealing with the same author or two authors who have identical names. Of course, it’s still possible for two authors to have different names that share the same 20 first letters.

This book is a hash table for the real numbers. It has 1020 drawers, but to save paper only the nonempty drawers are shown. But it differs very significantly from the card catalog of a library, in which the indexing terms (the “keys”) have finite length. When you examine two catalog cards, you can tell immediately whether the

es are very long. When you look up an entry in this book, you can’t tell immediately whether the expressions given are identically equal or not. That’s because their decimal expansions are of infinite length and all you know is that they share the same first 20 digits. With that warning, I can tell you there is no case in this book known to the author in which two listed quantities have the same initial 20 digits but are known not to be equal. There are many cases, however, in which the author has no idea whether two expressions for the same entry are equal, but therein lies the opportunity for much research by author and reader alike.

As a final example, seeing with some surprise that sin sink

k

k

kk k

1

2

21

1

2

led me to an

investigation of the conditions under which the sum of an infinite series equals the sum of the squares of its individual terms. (We may call such series “element-squaring.”) Element-squaring series rarely occur naturally but are in fact highly abundant:

Theorem 3. Every convergent series of reals B bkk

1

such that B bkk

( )2 2

1

can

be transformed into an element-squaring real series

B bk ing to B a single real by prependk 0

term b0 iff B B( )2 1 . Proof: see the end of the cha

I hope that these examples show that the list offers both information and challenges to the reader. It is a reference work, but also a storehouse of questions. Anyone who is fascinated by mathematics will be able to scan through the pages here entrie for a ber are related.

4

and wonder if, how and why the s given num

ts. The reader is encouraged to develop his own sup

pter.

The selection of entries in this book necessarily reflect the interests and prejudices of the author. This will explain the relatively large number of expressions involving the -function. For this I make no apology; there are more interesting real numbers than would fit in any book, even listing only their first 20 digi

plements to this catalog by using Mathematica to develop a list of numbers useful in his or her own field of study.

PREFACE vii

Professor George Hardy once related the now-famous story of his visit to an ailing Ramanujan in which Ramanujan asked him the number of the taxicab that brought him. Hardy thought the number, 1729, was a dull one, but Ramanujan instantly answered that it was the smallest integer that can be written as the sum of two cubes in two different ways, as 1 123 3 and

integer such that

9 103 3 . Hardy marveled that Ramanujan seemed to be on intimate terms with the integers and regarded them as his personal friends1. It is hoped that this book will help make the real numbers the reader’s personal friends.

Proofs of Theorems 1-3

Theorem 1. For a 1 an integer, b an a b 2 and c a positive real

num( )b

c k

aj

ckjj

a

1 1

1at t the chapter.

ber, then kb b . A proof is given he end of

Proof: The left-hand sum may be written as the double sum 1

11aj b j

kj

k c

order of summation,

. Reversing the

1111

11 11 1 k jk jj k

1

1jabjajbjbaj

ckkckkck , which equals

1 1

1

1

1 1k ck ck kbk

a a b bk

p

. QED.

Theorem 2. For c > 1 real and prime, 1

11 1c

kp

cpi

kpk

i

( ), where ( )n is the Moebius

function.

Proof: ( ) ( )kp kp

kp jkp

. Note thc ck jk 11 11

at every term of 1

i

1 c pi

sum, certainly in the case when

appears at least once in

the ouble 1 and j pik d . We will show that the sum of all

coefficients of 1

c pi in the double sum is zero except when k 1 is a power of p, in which and j

case ( )kp 1, and the result follows. Suppose that jk is a power of p. Then k ( )kpif 1, 0 since kp has a repeated factor.

Therefore, only the terms with k 1 contribute to the sum and ( )kp 1 each j a power of p. Now consider all pairs j, k with jk not a power of p. Suppose that i has the factorization i

p a1 th each of the a 0. pq , wi i If aq1 p k or if k has any repeated prime factors, then

( )kp 0. Since k i, it suffices to consider only those k whose factors are products of subsets P P o set f distinct primes taken from thek ppp q 1 , a set having cardinality q 1 or q,

on whether depending P contai If ns p. Pk as an odd number of elements, then h ( )kp 1; , if otherwise Pk has an even number of elements, then ( )kp 1. ( )kp 0 cannot be zero

since p does not divide k. But the number of subsets of Pk having an odd num

ber of elements

1This incident is partially related in Kanigel, R. The Man Who Knew Infinity. New York: Washington Square Press (1991). ISBN 0-671-75061-5. The book is a fascinating biography of Srinivasa Ramanujan.

PREFACE viii

is ing an even numbeble sum

the same as the number of subsets hav r of elements. Therefore, the terms cancel each other and terms with jk not a power of p do not contribute to the dou . QED.

Theorem 3. Every convergent series of reals B bkk

1

such that B bkk

( )2 2

1

can

be transformed into an element-squaring series

B bkk 0

by appending to B a single real term

b0 iff B B( )2 1

4 .

Proof: For B to be element-squaring, it is necessary that

b b b b B b Bkk

kk

2 2 2 2 2

( ) . Solving this quadratic equation for b yields 00

01

0

0

bB B

0

21 )1 4

( )(

2, which is real iff B B( )2 1

4 0 . Since is finite, b B b B0

that if

converges. It is also eleme y to showntar B B( )2 1

4 , then there is no real ele ent that m

can be prepended to B to m ke it element-squaring. QED.

a

Note that B does not necessarily imply that B( )2 b. For example, let kk

k

( )1

.

Then

2

1,

1B , which is finite, but 2

Bkk

( )2 1

1

, which ( )2Bdiverges. Likewise, does

not imply that . If bkk 1

, then B( )2

6

2 but B diverges. B

Using This Book

This book is a catalog of over 10,000 real numbers. A typical entry looks like:

.20273255405408219099... log log3 2

2

1

5

arctanh AS 4.5.64, J941, K148

= 1

5 2 1

1

22 10

1

11

kk

k

kkk k

( )

( )

=

12)12(

log

x

dxx

= dxx

xxx1

0

5sinsin)( GR 1.513.7

Each entry begins with a real number at the left, with its integer and fractional parts separated. Entries are aligned on the decimal point. Three dots are used to indicate either (1) a non-terminating expansion, as above, or (2) a rational number whose period is greater than 18. Three dots are always followed by the symbol , which denotes “approximately equal to.”

The center portion of an entry contains a list of expressions whose numerical values match the portion of the decimal expansion shown. The entries in the list are separated by the equal sign =; however, it is not to be inferred that the entries are identically equal, merely that their decimal expansions share the same prefix to the precision shown.

There is no precise ordering to the expressions in an entry. In general, closed forms are given first, followed by sums, products and then integrals. At the right margin next to expression may appear one or more citations to references relating the expression to the numerical value given or another expression in the entry. When more than one expression appears on a line containing a citation, it means that the citation refers to at least one expression on that line. See also, “Citations,” below.

Format of Numbers

Numbers are given to 19 decimal places in most cases. Where fewer than 15 digits are given, no more are known to the author. Rational numbers are specified by repeating the periodic portion two or more times, with the digits of the last repetition underlined. For example, 1/22 would be written as .04504545, which is an abbreviation for the non-terminating decimal .0454545454545045... . Since rational numbers can be specified exactly, the = symbol rather than is used to the right of the decimal. Where space does not permit a repetition of the periodic part, it is given once only. If the period is longer than 18 decimals, the sign is used to denote that the decimal has not been written exactly. No negative entries appear.

Ordering of Entries

Entries are in lexicographic order by decimal part. Entries having the same fractional part but different integer parts are listed in increasing magnitude by integer part. Rational numbers are treated as if their decimal expansions were fully elaborated. For example, these entries would appear in the following order:

USING THIS BOOK x

.505002034457717739... .5050150501 = .505015060150701508... .5050 = .505069928817260012...

Arrangement of Expressions

In general, terms in a multiplicative expression are listed in decreasing order of magnitude except that constants, including such factors as ( )1 k appear at the left. For example, 4 2 3k kk! is used in preference to 4 2 . Additive expressions are ordered to avoid initial minus signs, where possible. For example, 3

3k kk !2 k is used instead of 2 3k . These principles are

applied separately in the numerators and denominators of fractions.

Citations Abbreviations flushed against the right margin next to an expression denote a reference to

an equation, page or section of a reference work. Page numbers are prefixed by the letter p, as in “p. 434.” A list of cited works appears at the end of the book.

Acknowledgments

No book like this one has previously appeared, probably because of the monumental difficulty of performing the required calculations even with computer assistance. However, with the development of the Mathematica® program, a product of Wolfram Research, Inc., it has been possible to obtain 20-digit expansions of most of the entries without undue difficulty. The symbolic summation capability of that program has been invaluable in developing closed-form expressions for many of the sums and products listed. Even though my involvement with computers extends back almost 35 years, I have no hesitation in declaring that Mathematica is the most important piece of mathematical software that has ever been written and my debt to its authors is, paradoxically, incalculable.

The tendency to be fascinated with numbers, as opposed to other, abstract structures of mathematics, is not uncommon among mathematicians, though to be afflicted to the same degree as this author may be unusual. If so, it because of the role played by a collection of people and books in my early life. My grandfather, Max Shamos, was a surveyor who, to the end of his 93-year life, performed elaborate calculations manually using tables of seven-place logarithms. The work never bored him — he took an infectious pride in obtaining precise results. When calculators were developed (and, later, computers), he never used them, not out of stubbornness or a refusal to embrace new technology, but because, like a cabinetmaker, he needed to use his hands.

I read later in Courant and Robbins’ What is Mathematics, of the accomplishments of the calculating prodigies like Johann Dase and of William Shanks, who devoted decades to a manual determination of to 707 decimal places (526 of them correct). At the time, I couldn’t imagine why someone would do that, but I later understood that he did it because he just had to know what lay out there. I expect that people will ask the same question about my compiling this book, and the answer is the same. The challenge set by Shanks is still alive and well — every year brings a story of a new calculation — is now known to several billion digits.

My father, Morris Shamos, is an experimental physicist with a healthy knowledge of mathematics. When I was in high school, I was puzzled by the formula for the moment of

inertia of a sphere, M m r2 . That is was proportional to the square of the radius made

sense enough, but I couldn’t understand the factor of 2/5. Dad set me straight on the road to advanced calculus by showing me how to calculate the moment by evaluating a triple integral.

2

5

As a teenager during the 1960s, I worked for Jacob T. (“Jake,” later “Jack”) Schwartz at the New York University Computer Center, programming the IBM 7094. (That machine, which cost millions, had a main memory equivalent to about 125,000 bytes, about one-tenth of the capacity of a single floppy disk that today can be purchased for a dollar.) NYU had a large filing cabinet with punched-cards containing FORTRAN subroutines for calculating the values of numerous functions, such as the double-precision (in the parlance of the day) arctangent. By poring over these cards, I learned something of the trouble even a computer must go through in its calculations of mathematical functions. It was this exposure that drew me to field that later came to be called computer science.

ACKNOWLEDGMENTS xii

For the past 20 years I have been an enthusiastic user of Neil Sloane’s A Handbook of Integer Sequences in combinatorial research, so it is particularly pleasing to note that this book itself consists entirely of such sequences.

My greatest debt, however, is to Stephen Wolfram for his creation of Mathematica, the single greatest piece of software ever written. In developing this catalog, I used Mathematica on and off over a 15-year period to explore symbolic sums and integrals and to compute most of the constants that appear in this book to 20 decimal places.

Notation

a n( ) greatest odd divisor of integer n.

Ai x( ) Airy function, solution to y xy 0. AS 10.4.1

b x( ) ( )

1

0

k

k x k.

Bk kth Bernoulli number, B Bk k ( )0 . AS 23.1.2

B xk ( ) kth Bernoulli polynomial, defined by te

eB x

t

k

xt

t k

k

k

1 0

( )!. AS 23.1.1

bp n( ) number of divisors of n that are primes or powers of primes.

ck coefficient of xk in the expansion of ( )x .

C x( )

0

2

142

0

2

)14(2)!2(

)1(cos

2

1)(

kk

kkkx

kk

xdttxC

. AS 7.3.1, 7.3.11

Ci x( ) cosine integral,

x

k

kk

kk

xxdt

t

txxCi

0 1

2

)2()!2(

)1(log

1coslog)( .

AS 5.2.2, AS 5.2.16 d n( ) number of divisors of n, d n n( ) ( ) 0

dn kth derangement number, d d d d1 2 3 40 1 2 9 , , , , nearest integer to k e!/ . Riordan 3.5

E x( ) complete elliptic integral

dxxE 2/

0

2sin1)( .

En nth Euler number,

2

12 k

kk EE . AS 23.1.2

E xn( ) nth Euler number, defined by 2

1 0

e

eE x

t

k

xt

t k

k

k

( )!

. AS 23.1.1

Ei x( ) exponential integral

1 !

log)(k

k

x

t

kk

xx

t

dtexEi . AS 5.5.1

Erf x( ) error function

0

12

0

12

0 !)!12(

2)1(2

)12(!

)1(22 22

k

kkkx

k

kkxt

k

xe

kk

xdte

. AS 7.1.5

Erfc x( ) complimentary error function = 1 Erf x( ). AS 7.1.2 f n( ) number of representations of n as an ordered product of factors

Fn nth Fibonacci number, F F F F Fn n n1 2 11 2 ;

0 1F (; ; )a x hypergeometric function x

k a

k

kk ! ( )

0

, where a a a a kk( ) ( )...( ) 1 . Wolfram A.3.9

1 1F ( ; ; )a b x Kummer confluent hypergeometric function x a

k b

kk

kk

( )

( )!

0 ,

NOTATION 599

where a a a a kk( ) ( )...( ) 1 . Wolfram A.3.9

2 1F ( ; ; ; )a b c x hypergeometric function x a b

k c

kk k

kk

( ) ( )

( )!

0

, where a a a a kk( ) ( )...( ) 1 . Wolfram A.3.9

Catalan’s constant ( )

( ). ...

1

2 10 915965594177219015052

0

k

k k G

Gn

1 )13(

1

)13(

11

knn kk

gn

1 )13(

1

)13(

11

knn kk

gd z( ) Gudermannian, gd z ez( ) arctan 22

Hn harmonic number: Hkn

k

n

1

1

H ne even harmonic sum: H

ke

n

k

n

1

21

H no odd harmonic sum: H

ko

n

k

n

1

2 11

j( )n generalized harmonic number: H

kj

n

jjj

n jk

n j j( )

( )

( )( ) ( )

( )!,

1 1 1

11

1

1 1

H

hn

0 )56(

)1(

)16(

)1(

kn

k

n

k

kk

hg x( ) Ramanujan’s generalization of the harmonic numbers, hg xx

k k xk

( )( )

1

I xn( ) modified Bessel function of the first kind, which satisfies the differential equation

z y zy z n y2 2 2 0 ( ) . Wolfram A.3.9

jn

1 )13(

1

)13(

11

knn kk

J311

J xn( ) Bessel function of the first kind ( )

!( )!

1

2

2

20

k n k

n kk

x

k n k LY 6.579

k 1 2 k

K x( ) complete elliptic integral of the the first kind

dx

2/

02sin1

1

K xn( ) modified Bessel function of the second kind, which satisfies the differential equation

z y zy z n y2 2 2 0 ( ) . Wolfram A.3.9

l x( )

n

kk k

k

xk

xk

k

k

11

log)log(log , when x is an integer n.

NOTATION 600

li x( ) logarithmic integral x

t

dtxli

0 log)(

Li xn( ) polylog function Li xx

kn

k

nk

( )

1

log x natural logarithm loge x

a prime number. denotes a sum over all primes. pp

pd n( ) product of divisors pd n dd n

( )|

pf n( ) partial factorial pf nkk

n

( )!

1

0

pfac n( ) product of primes not exceeding n, pfac n pp n

( )

pq n( ) product of quadratfrei divisors of n, pq n pr n( ) ( ) 2

pr n( ) number of prime factors of n

r n( ) 4 ( )|

dd n HW 17.9

s n( ) smallest prime factor of n

sd n( ) sum of the divisors of n, sd n n( ) ( ) 1

S x( ) 34

1

2

0

2

)34()!12(

)2/()1(

2sin)(

k

k

kkx

xkk

dtt

xS

. AS7.3.2, AS 7.3.13

S n m1( , ) Stirling number of the first kind, defined by the recurrence relation:

x x x n S n m xm

m

n

( )...( ) ( , ) 1 1 1

0

. AS 24.1.3

S n m( , )2 Stirling number of the first kind, nm

k

km kk

m

mmnS

02 )1(

!

1),( . AS 24.1.4

Si x( ) sine integral,

x

k

kk

kk

xdt

t

txSi

0 0

12

)12()!12(

)1(sin)( . AS 5.2.1, AS 5.2.14

T s( ) 1

ksk , where k has an odd number of prime factors

v1

2

23)31(

2

23)31(1

ii

iiv

w x( ) w x e erfc ixix

kx

k

k

( ) ( )( )

( / )

2

2 10

Wn

0 )56(

)1(

)16(

)1(

kn

k

n

k

kk

z x2 ( ) z x k xx

x xk

k2

2

12

11( ) ( ( ) ) ( ( ))

NOTATION 601

( )s ( )( )

( )s

k

k

sk

1

2 1

1

0

Euler's constant

n k

n

knlim log

1

1

0.57721566490153286 AS 6.1.3

f

n

kn

dxxfkf1 1

)()(lim

( )s Riemann zeta function 1

1 ksk

( )m ( )s derivative of the Riemann zeta function ( log )

k

k

m

sk 1

( )s ( )( )

( )sk

sk

s( )

k

s

1

1 21

1

1

( )s ( )( )

( ) (sk

ss

)k

s

1

2 11 2

0

( )n Moebius function, ( ) , 1 1 ( ) ( )n k 1 if n has exactly k distinct prime factors, 0 otherwise. ( )n number of different prime factors of n.

k n( ) sum of kth powers of divisors of n, dk

d n|

golden ratio, 1 5

21 61803397498948482

. ...

( )n Euler totient function, the number of positive interegs less than and relatively prime to n.

( )n polygamma function ( )nkk

n

1

1

1

( )x kth derivative of the polygamma function ( ) ( )( )k

k

kx

d x

dx

( )n ( ) ( )n kk

n

1

( )s )()1(

21)( 2/ ss

ss s

( )n overcounting function ( )n -1 + the number of divisors of the g.c.d of the exponents of the prime factorization of n.

.00000000000000000000 = ( ) ( )

!

( )

!

1 1 1

0

1 2

1

k

k

k

k

k

k

k

k GR 1.212

= ( ) ( )

!

( )

!!( )

1 2 2 11

2

20 1

2

1

k k

k

k

k

kk

k

k

k

k

k

k

k

k

= ( )k

kk

1

Berndt Ch. 24 (14.3)

= ( )k

kk 2 11

= ( )sin

1 13

1

k

k

k

k

= log2

20 1

x

x

1 .000000000000000000000 = 1

11 k kk ( )

J396

= 1

2 21 kk

½

= k

kk ( ) !

11

GR 0.245.4

= 2

20

k

k

k

k( ) !

= 1

10 ( )!k k!k

= 2 1 3 3 1

3 3

4 3

72

2

6 5 41

k k

k k

k k

k k k k3k k

= 2 1

4 1

2 1

2 11 1

k

k k

k

k kkk k

( )

( )!

( )! ( )

!

=

( ) logk k

kk 1

= F k

kk

4 2

1 8

=

22 22k

k

kk

H

= ( )kk

12

=

2

)(

k k

k

=

( ) logk k

kk 1

Berndt Ch. 24, Eq. 14.3

= 1

1

S

, where S is the set of all non-trivial integer powers

(Goldbach, 1729) JAMA 134, 129-140(1988)

=1

21 n

k

k

n

kn !

log

!

= ( )sin

121

1

k

k

k

k

= 11 1

1

( )k

k k J1065

= dx

x

x dx

x21

21

log

= dx

x( ) /2 30 1

2

= dx

x x( )1 1 20

1

= x x

edxx

log( )1

0

=

cossin

xx

x

dx

x0

Berndt I, p. 318

= arctan x

xdx2

1

2 .000000000000000000000 = (½, , )!

( )! ( )!

!

1 02

2

2 11 1 1

k k

k

k

kkk

k

k k

= ( )

( )

k H F

k

k

kk

kk

kk

kk

k

1

2 2

1

4 1

2

1 1 0

=

1

1)2(exp

k k

k

=

12

)(

k k

kz

= 11

2 11

2

22

k k

k

kk k( )

= 11

220

k

k

= e ek kk

k

k

k

( ) / /

1

1

1 2

1

1

= log

(cos )

2

21

2

20

2

1

x

xdx

x

edx

dx

o

= x x

xdx

sin

cos

/

10

2

GR 3.791.1

= logsin cos/

x x d2

0

2

x

= e dxx

0

1

=

0

11

3xe

dxx

3 .000000000000000000000 = k k

kkk 6

2 1

0

4 .000000000000000000000 = k k k

k

k

kkk

kk

kk2

1

2

1

4 1

2 21

1 2 0

( )

( )

= ( )!!

( )!! ( )

( )!!

( )!! (

2

2 1

2

2 122

121

k

k k k

k

k k kk k

)

6 .000000000000000000000 = (½, , )

2 02

2

1

kk

k

= k

k kk

2

3 2 2( )( )

= x dx

e

xdx

xx

3

0

3

21

log

10 .000000000000000000000 = ( )k F k

kk

kk

k

4

2 20 1

12 . 000000000000000000000 = ( )

log

k dx

x xxe dxk

k

x

1

2

2

0 2 0

15 . 000000000000000000000 = ( / , , )1 3 4 03

4

1

kk

k

20 . 000000000000000000000 = k

k kk

2

4 3 3( )( )

24 . 000000000000000000000 = log4

20

xdx

x

26 . 000000000000000000000 = (½, , )

3 02

3

1

kk

k

52 . 000000000000000000000 = ( )k

kk

1

2

3

0

70 . 000000000000000000000 = k

k kk

2

5 4 4( )( )

94 . 000000000000000000000 = F kk

kk

2

1 2

150 . 000000000000000000000 = (½, , )

4 02

4

1

kk

k

1082 . 000000000000000000000 = (½, , )

5 02

5

1

kk

k

1330 . 000000000000000000000 = F kk

kk

3

1 2

25102 .000000000000000000000 = F kk

kk

4

1 2

1 .00000027557319224027... 1

101 ( )k !k

... 697

6613488

8

8

W J313

...

648000, the number of radians in one second

1 .000006974709035616233... 2coth

... 10

1 .0000115515612717522... 547 61

2 3 5

1

6 5

1

6 1

7

10 9

1

7

1

71

( )

( )

( )

( )

k k

k k k J328

1 .000014085447167... W7 J313

.0000248015873015873... 13

1 ( )k

!k

... 9

... sec1

.00004439438838973293... 1

22528

11

121

23

12

1

4096 12 1 122

Fdx

x, , ,

... e10

1 .0000513451838437726... ( )

( )

( )9

511 9

512

1

2 1 90

kk

AS 23.2.20

1 .00005516079486946347... 1

358318080

1

12

5

12

7

12

11

125 5 5 5 ( ) ( ) ( ) ( )

= ( )

( )

( )

( )

1

6 5

1

6 1

1

6

1

61

k k

k k k

1 .0000733495124908981... W6

691

87480

J313

... 8

... e9

...

( ) ( )log3

3

21

2k

kk

1 .000155179025296119303... ( )

( )

( )8

17

161280

255 8

256

1

2 1

8

80

kk

AS 23.2.20

1 .0002024735580587153... 3

31

.0002480158734938207... 1

81 ( )k !k

...

1

8192

1

16

1

16 12

30

( )

( )kk

.0002499644008724365... 16 6 112 6

2 1k kk

k k

( )

1 .0002572430074464511... 5 61

2 3

1

6 5

1

6 1

7

7 6

1

5

1

51

( )

( )

( )

( )

k k

k k k J328

.00026041666666666666 = 1

3840

1

5 25

!

.0002908882086657216...

10800, the number of radians in one minute of arc.

.000291375291375291375 =1

34321

492

8

kk

...

( )

( )( )3

4096

1

81921

1

162

31

kk

... 7

... e8 1 .00038992014989... W5 J313

1 .000443474605655004... ( )

( )

( )

( )

1

4 3

1

4 1

1

7

1

71

k k

k k k J326

1 .0004642141285359797... 1

7776

1

6

1

6 13

41

( )kk

1 .00047154865237655476... ( )

( )

( )7

127 7

128

1

2 1 70

kk

AS 23.2.20

1 .00049418860411946456... ( )11111

1

kk

1 .00056102267483432351...

1

3456

1

12

1

12 112

31

( )

( )kk

1 .000588927172046... r( )16 Berndt 8.14.1

5

82

1 2

4 2

2 2

16

2 2 2

2 2 2

2 2

16

2 2 2

2 2 2log

log( )log log

... 11

22

21

24

9

23

15

25

1

2

1 3

51

( ) ( ) ( ) ( )( )

( )

k

k

k

k

.00069563478192106151... ( )

( )

3

1728

1

12 31

kk

... ( )( )

12

21

2

1

kk

k

k k

.00088110828277842903...

1

3456

11

12

1

12 12

31

( )

( )kk

.00091188196555451621... e7

1 .00097663219262887431... 15

1 k kk

1 .00099457512781808534... ( )10110

1

kk

.00099553002208977438... ( )10 11

kk

.00101009962219514182... 13

124 6 4 1

1

110 4

2

( ) ( )kk kk k

.00102614809103260947... ( )5 5 11

110 5

2

kk kk k

.001040161473295852296... 6

.0010598949016150109... 15

8 42 6 4 6 1

1

110 6

2

coth ( ) ( ) ( )k

k kk k

.00108225108225108225 =1

9241

362

7

kk

.00114484089113910728...

1

3456

5

6

1

12 22

31

( )

( )kk

1 .00115374373790910569... 1

124416

1

12

5

12

7

12

11

123 3 3 3 ( ) ( ) ( ) ( )

=( )

( )

( )

( )

1

6 5

1

6 1

1

4

1

41

k k

k k k

1 .001242978120195538...

1

1458

1

9

1

9 82

31

( )

( )kk

1 .00130144245443407196... 1

31457280

1

8

3

8

5

8

7

85 5 5 5 ( ) ( ) ( ) ( )

=( )

( )

( )

( )

1

4 3

1

4 1

1

6

1

61

k k

k k k

.001311794958417665... 197

27 64

3 3

32

1

2 8 30

( ) ( )

( )

k

k k

.00138916447355203054... e

e kk4

1

41

1

2

1

4 21

cos

( )!

3 .00139933814196238175... 2

1 .0013996605974732513... 3 3

62

3

4

3 1

312

log

log log( )( )r Bendt 8.14.1

.00142506011172369955... 1

102 k kk log

.00144498563598179575... ( )2

62

k

kk

.00144687406599163371... ( )2

162

k

kk

1 .0014470766409421219... 6

61960

663 6

64

1

2 1

( )( )

( )kk

AS 23.2.20

...

1

8192

9

16

1

16 92

30

( )

( )kk

... ( ) ( )log4

4

21

2

k

kk

.00153432674083843739...

1

3456

3

4

1

12 32

31

( )

( )kk

.0015383430770987471... 930 5

1920 2 1

6

61

( )

( )

k

kk

.0015383956660985045... 4 3

256

1

4 8 30

( ) ( )

( )

k

k k

1 .001712244255991878...

1

1024

1

8

1

8 72

31

( )

( )kk

1 .00171408596632857117... 5 3

6

( )

... 1

1536

1

4

1

4 13

40

( )

( )

kk

.001815222323088761... 5

54

197

216

1

1 2 3

2

2 2 21

2

k k k kk ( ) ( ) ( ) LY 6.21

.001871372759366027379... 7

360

3

2

1

1

3

3 21

( )

k e kk

Ramanujan

=

32

1 1

( )k

e kk

.001871809161652456210... 1

12 21

22

1k e

k

ekk

kk

( )

.001872682449768546116... 1

121

12

1k e

k

ekk

kk

( )

.00187443047777494092... 1

1 121

02

1e

k

ekk

kk

( )

.001912334879124743... 15 109

960

2

6

2

0

e

k

k

k

( )!

1 .001949804343... j9 J311 1 .00195748553... J309 G9

2 .001989185473323053... 1110

1

kk

1 .00200839282608221442... ( )9

.00201221758463940331... ( )9 11

119

2

kkk k

.00201403420628152655... 3

46

3

25

1

23

5

44

9

83

17

162

387

642 ( ) ( ) ( ) ( ) ( ) ( )

= H

kk

k

15

1 2( )

.00201605994409566126... ( )8 1 11

19

2

kk kk k

.00201681670359358... 3277

3375 32

1

2 7

3

30

( )

( )

k

k k

.00202379524407751486... ( )7 2 11

19 2

2

kk kk k

.002037712107418497211... )7()3()5()10(4

7)1,9( 2 MHS

.00203946556435117678... ( )6 3 11

19 3

2

kk kk k

... 7 3

4096

1

8192

1

2

1

16 82

30

( )

( )( )

kk

.00210716107047916356... ( )5 4 11

19 4

2

kk kk k

.0021368161356763007...

1

3456

2

3

1

12 42

31

( )

( )kk

.00213925209462515473... ( )4 5 11

19 5

2

kk kk k

1 .0021511423251279551... 5

486

4

4

W J313

... 19

11520

2

96

2

48

3 2

8

5 3 2 4

log log ( ) log

=1

4 2 1

25

0k

k k

k

k( )

.00227204396400239669...

1

1458

8

9

1

9 12

31

( )

( )kk

1 .00277832894710406108... cosh cos1 1

.00228942996522361328... 1

3 6 19 6

1 1k kk

k k

( ( ) )

.00234776738898358259... ( )

( )

3

512

1

8 31

kk

... log

logsin/2

4 2 0

4

G

x dx GR 4.224.2

=

log/ x

xdx

1 20

1 2

GR 2.241.6

1 .0024250560555062466... 2 2

5

5

4

3

2 5

1 5

210

log loglog ( )

r Berndt 8.14.1

.002450822732290039104...

dx

e

exx

x

2

4

5 14

3

.00245775047043566779... 171109

24 5

2

1

ek

kk

( )!

.002478752176666358... e6

2 .0025071947052832538... 6 2 8 2 62 2

22

0

log log( )

k H

kk

kk

.00260416161616161616 =1

284

1

4 24

!

... 1

2

1

2

1

2

1

4 4

1

2

1

2

1

4 160

cos cosh cos( )!

e

e

k kk

... 1

8

5

3

1

4

1

4 7 12

log arctan

dx

x x

... dx

x72 1

... loglog

263

6 72

dx

x x

... dx

x x7 22

... 1

4

5

3

1

8 7 32

log

dx

x x

... cosh cos1 1

... loglog

27

3

1

24 7 42

dx

x x

.00286715218149708931... 1

92 k kk log

...

1

3

1

)5(

)1(

4

)3(3

216000

195353

k

k

k

=

1

0

1

0

1

0

555

1dzdydx

xyz

zyx

5 .0029142333248880669... 2

10

k

k k !

.002928727553540092... ( ) ( )

( )

3

36 324

1

3

2 1

31

k

k

k

k

...

1

8192

7

16

1

16 72

30

( )

( )kk

.0031244926731183736...

1

3450

7

12

1

12 52

31

( )

( )kk

.00318531972162645303... 2

33

2 33 2

2ex

x x

sinsin

... loglog

23

2

9

24 7 52

dx

x x

.00326776364305338547... 5

...

1

1458

7

9

1

9 22

31

( )

( )kk

... Re ( ) i

.0033104481564221302... 1

16

7

45 256

4

.0033178346712119643... 3 3

32

7

64

1

2 6 30

( ) ( )

( )

k

k k

=

1

0

1

0

1

0222

555

1dzdydx

zyx

zyx

24 .00333297476905227... e e e e

e e

k

ekk

( )

( ), ,

3 2

5

4

1

11 11 1

1

14 0

... 15 5

16

31

32

1

3 50

( ) ( )

( )

k

k k

.00337787815787790335...

1

1024

7

8

1

8 12

31

( )

( )kk

.003472222222222222222 =1

288 1 2 3 4 51

k

k k k k kk ( )( )( )( )( )

.00364656750482608467... 1

27

3

36

1

3 3

1

31

( ) ( )

( )

k

K k

=

1

0

1

0

1

0333

555

1dzdydx

zyx

zyx

...

3

2

36 3

91 3

216

1

432

1

6

( ) ( ) Berndt 7.12.3

= 1

6 5 31 ( )kk

.0037145966378051377... log( log( log(log ))) 2 1 .00372150430231012927... log( log(log ))2

.00374187319732128820... coth

12

12e

1 .00374187319732128820... coth

e e

e e

=3

800

1

2 3 41

5

( )( )( )(k k k k )k

... 19

2 2

1

4 3

1

4 1

5

12

1

51

1

5

( )

( )

( )

( )

k

k

k

k k J 326

1 .003795666917135695...

4 3 2 5

8 2 1 40

2arctan log dx

x

.00388173171751883... 32

256

1

4 2

3 1

31

( )

( )

k

k k

... 22

72

2

3

1

2 1

3

0

log( )

( )

k

kk

k

k

1 .0038848218538872141... arctan2

... j8 J 311

... 14

1 k kk

... 656

6200145

8

8

G J 309

.00396825396825396825 =1

2525 1

252

6

( )kk

.00402556575171262686... 7 4

8

7

8 4

18 4

2

( )

sinh

( )

k

k k k

=( ) ( ) ( )

int

1 1

142

4

1

k

k a non trivegerpower

k

k

ial

.004031441804149936148...

02

2

3 18

1dx

e

exx

x

.00406079887448485317... 19450 8 1

88

( )

( )

1 .0040773561979443394...

8

94508 ( )

.00409269829928628731... 15

16 8

2

8

2 2

2 2

cothsin sinh

cos cosh

= ( )8 11

118

2

kkk k

... 10

... 17 1 18

2 1k kk

k k

( )

... 3

42

2 3

3

2

( ) tanh

= 16 2 18 2

2 1k kk

k k

( )

.004147210828822741353... 1

12 52 ( )kk

... dxx

x

1

0

22

1

arccos

2421616

.004166666666666666666 =1

2407 ( )

.004170242045482641209... )6()3()5()4()7()2()9(4)1,8( MHS

... 15 3 18 3

2 1k kk

k k

( )

... 15

84

4 ( ) coth

= 14 4 18 4

2 14

1k kk

k

kk k k

( )( )

.00442764193431403234... ( )

( )( )

( )

2

2

7

1803 3 1

1

2

4 1

41

2

k

k

k

k

... 1

486

1

3

1

3 13

40

( )

( )

kk

.00450339052463230128... ( )2

52

k

kk

.00452050160996510208... ( )2

152

k

kk

...

2

2

22 2

2

82 2 1 2

cos sin csc

= k kk

kk

k k

2 2 12

21

2

2 22

( )( )

... 31 5

325

1

2 1 50

( )( )

( )

kk

AS 23.2.20

... 3

256

1

9

1

2 1 2 1 2 3

2

2 21

2

( ) ( ) (k k k )k

LY 6.21

... 12

3

64

1

2 3

3 1

31

G k

k

k

k

( )

( )

... log2661

960 7 62

dx

x x

... 13 5 18 5

2 1k kk

k k

( )

...

1

8192

3

8

1

16 62

30

( )

( )kk

... Gx

xdx

log log( )2

2

1

1

2

20

1

GR 4.295.6

=

log sin

/ xdx

20

2

... log log ( )22

41 2 8 r

.0047648219275670510... 67

64 96 360

3

8

5

2

1

2

2 4

52

( ) ( )

( )k kk

... 7

216

1

6 5

1

6 1

3 1

31

1

3

( )

( )

( )

( )

k

k

k

k k

... 7 3

1728

1

12 6 31

( )

( )

kk

.004915509943581366... sin

2

2 4

2

22

1

kk

.004983200... mil/sec

.00506504565493641652...

1

1458

2

3

1

9 32

31

( )

( )kk

.00507713440452621921... 4

51

93 5

192 2 1

( )

( )

k

kk

... 4

3

10 3

81

=kk

k

F Fk

2

12 1 2 12

1

22 2

3

2

1

4

1

33 3

5

2

1

4

, , , , , ,

...

1

3

1

)4(

)1(

1728

1549

4

)3(3

k

k

k

=

1

0

1

0

1

0

444

1dzdydx

xyz

zyx

.0051783527503297262... 7 3

128 512

1

1024

3

4

1

8 2

32

31

( )

( )( )

kk

.0053996374561862323... 15

42 4 6 2 6

1

( ) ( ) ( ) ( )kk

1

=1

8 62 k kk

=1

180 2 3 4 51

k

k k k kk ( )( )( )( )

6 .005561414313810887... 2

2 23

2

0log

x

dxx

... ( , )5 3

1 .00570213235367585932... 1

2

2

3

3

4

1

216 631

log log

k kk2 Ramanujan] Berndt Ch. 2

... 1

81 k kk log

.005899759143515937... 45 7

8 6

( )

Berndt 9.27.12

1 .005912144457743732... 1

250

1

5

1

5 42

31

( )

( )

kk

1 .0059138090647037913...

3

1 33

243

91 3

486

1

324

1

6

1

17496

1

6

( ) ( ) ( )

=( )

( )

2 1

6 1

2

40

k

kk

... 3

3032

26

27

1

2 5

( )

( )

k

k k

.00603351696087563779... log k

k 7

...

1

4096

7

162

2

8 81

( ) log x

x xdx

...

1

3375

7

152

2

8 71

( ) log x

x xdx

...

( ) log( )3

196

1

2744

1

22

2

8 61

x

x xdx

...

2 )(

)1(

)1(

)(

k k

k

k

k

... k

kk2

1 1

...

1

2197

7

132

2

8 51

( ) log x

x xdx

...

1

1728

7

122

2

8 41

( ) log x

x x

...

1

42

2

16

5 2 2

5 2 2arctan log

( )

arctan arctanlog1 2

8

2

81 2 1 2

3

8

2

16 6 22

dx

x x

31 .0062766802998201758... 3 2

0 1

log( )

xx

x xdx GR 4.261.10

= x e

edx

x

x

2 2

1

/

GR 3.4.11.16

.00628052611482317663... 2 1

3148

2

8

3 3

32

1

2 2

2

log ( ) ( )

( )

k

k

k

k

00629484324054357244... 1

160

5

61

11

6

1

64 12 1 62

Fdx

x, , ,

...

1

1331

7

112

2

8 31

( ) log x

x xdx

... log log21

531 6

2

dx

x x

= 4

6251

41

3

1

( )kk

k

k

...

1

1000

7

102

2

8 21

( ) log x

x x

.006476876667430481... log arctan3

4

2

2

1

2 4 6 22

dx

x x

... e e e

e e

k

ekk

2

4

3

0

4 1

1

13 0

( )

, ,

... 5

3

...

1

729

7

92

2

8 11

( ) log x

x xdx

... ( ) ( )i i

... log log

( )

2

4

1

6 1 50

1

x

xdx

...

( )2 1

2 11

kk

1

k

... e5

...

1

512

7

8 12

2

81

( ) log x

xdx

... ( )( )

( )4 1

12

422

24

1

jk

f k

kji k

.00677885613384857755... 3

46 5

1

23 4 3 22 ( ) ( ) ( ) ( ) ( ) ( ) 5

= H

kk

k

15

1 1( )

... log

arctan7

6

1

3

5

3

3

6

1

8 6 32

dx

x x

... 2

1

28

11

108

arctan

...

962 2 8 2 12 3

1

4 2 1

22 3

40

log log ( )( )

kk k

k

k

... 2 3

343

2

81

( ) log

x

x xdx

.00702160271746929417... 7

48 180

3 3

4

23

8

1

1 2

2 4

41

( )

( )( )k k kk

... 16

3

128 1

3 2

2 31

Gx

xdx

log

( )

... 9

... 3

8

1 1

4

1

1

e k

k

k

( )

( )!

1

... ( ) ( )

1 21 4

1

k

k

k

1 .007160669571457955175... 1

230 kk

... 1

2454 3 2 452 ( )

.0071918833558263656... i i ii i cos( log ) sin( log )

...

1

216

7

62

8 21

( ) dx

x x

2 ...

1

216

1

62

1

216

7

6 12 2

2

60

1

( ) ( ) log x dx

x

... sech2

1

( )kk

... 17

360

3 3

4

1

2

4 1

41

( ) ( )

( )

k

k

k

k

... Gk

k

k

10016

11025

1

2 7

1

21

( )

( )

... 1

6

1

22

1

csch ( )k

k

[Ramanujan] Berndt Ch. 26

= 1

1329 ( )

.00763947766738817903... log3

2

13

24 6 42

dx

x x

... j7 J 311

... 1

71 ( !)kk

... 1716 15428

15

2

135

1

12

4 6

7 71

k kk ( )

... 119

72 6

1

1 2

2

2 21

3

k k k kk ( ) ( ) ( )

1 .00788821... J 309 G7

...

1

125

7

52

2

8 31

( ) log x

x xdx

... 3 5

44

4 4

( )( ) c Berndt 9.27.12

.008062883608299872296...

dx

e

exx

x

2

2

3 14

1

.00808459936222125346... 4

1 22

2

0

2

log logx e x dxx

...

1

8192

5

16

1

16 52

30

( )

( )kk

2 .00815605449274531515... 11 1 1 14

1 8 3 8 5 8 7 8

sin( ( ) sin( ( ) sin( ( ) sin( ( )/ / / /

= 11

81

kk

... 1

614496 1536

1

4

3

43 3 3

G ( ) ( )

... ( )( )

3 13 3

32

f k

kk

Titchmarsh 1.2.14

...

1

3456

5

12

1

12 52

30

( )

( )kk

.00828014416155568957... ( )

( )

7 1

7

... 1

2

1

3

2

3

3

2 3

1

3 2

1

1

e

e

k

k

k

cos( )

( )

!

1 .00833608922584898177... sin sinh sin

( )

1 1

2

1

2 4

1

4

1

4 10 !

e

e kk

... 1

81 1

1

8 41

cos sin( )

e

k

kk 1 !

1 .00834927738192282684... ( )7

=

0

624

)42)(32)(22)(12(4

)2(

381

64

381

)5(64

1143

)3(8

kk kkkk

k

=

0

236

)72)(62)(52)(32)(12(4

)2()1175793942604248(

7560 kk kkkkk

kkkk

Cvijović 1997

... ( )8 1 11

17 1

2

kk kk k

.008650529099561105501... )11(2

331)9()2(84)7()4(21)6()5(4)4,7( MHS

... 4

... ( ) log( )6 1 1 1

1

6

2

k

k

k

kk k

... 1

4

2

3

1

3

1 3

2

1 3

2

i i

= 16 1 17

2 1k kk

k k

( )

.0085182264132904860...

1

1458

5

9

1

9 42

31

( )

( )kk

... 15 2 17 2

2 1k kk

k k

( )

.008650529099561105501... )5()3(7560

)1,7(8

MHS

.00866072400601531257...

1

1024

5

8

1

8 32

31

( )

( )kk

...

1

64

7

4

1

64

3

4

2

272 2

2

8 41

( ) ( ) log x

x xdx

... 7

83

2

1

4

( ) ( )i i

= 13 4 17 4

2 1k kk

k k

( )

1 .00891931476945941517...

( )

( )

( )6

7 71

k

kk

Titchmarsh 1.2.13

5 .00898008076228346631... 1

20 ( )!kk

.00898329102112942789... i i3

... 91 3

216 36 3

1

432

5

6

1

6 1

32

31

( )

( )( )

kk

.009197703611557574338... 3

8 4 3 3 3

4 1

91

cot coth

( )kk

k

...

02 )7)(4(!

2)1(

4

7

4

52

k

kk

kkke

... 3 3 23

1 12

21

2

31

( ) log ( )

( )

k

k

k

k

... 5

3

2

3 90

1

3

3 3

2

3 3

2

4

i i

= 13 4 17 4

2 1k kk

k k

( )

... 7

45 16

15

16

7 4

8

15

16

1

3

4

40

( ) ( )

( )

k

k k

2 .00956571464674566328...

161

4

643

41

22

40

1Gx dx

x

log

...

96 2 2

1

8 7 8 5 8 3 8 11

( )( )( )(k k k kk )

J243

.00960000000000000000 = 6

625

3

61

log x

xdx

... ( )

( )

3

125

1

5 31

kk

... 64

9

8

)3(

=

1

0

1

0

1

0222

555

1dzdydx

zyx

zyx

... 7

108 1

3 2

60

log x

xdx

... 5

2

... 9 3

961

2 2

21

31

( )( )

( )

k

k

k

k

.0099992027408679778... 1

3072

1

4

3

4 63 3

3

( ) ( )

4

... 2

33

1

6

1

320

csch

( )k

k k

... i i i i2

4 2 2 21 1 2 2 ( ) ( )

i

2

2

2 2coth

=H

kk

k2 1

21

... 3 3

2

1549

864

2

6 51

( ) log

x

x xdx

... 4

...

1

27

7

32

2

8 51

( ) log x

x xdx

... 69

16

3

8

3

2

1

1 2

2

2 31

( )

( ) ( )k k kk

... 7 3

16 64

1

128

1

4

32

( ) ( )

Berndt 7.12.3

=

1

1024

1

8

5

8

1

4 32 2

31

( ) ( )

( )kk

.01037718999605679651... ( )k

kLi

k kk k

1 16

26

2

1

... 11

2

1

2

1

4 4

1

1

cos cosh( )

( )!

k

kk k

GR 1.413.2

.01041666666666666666 = 1

96 1 2 3 4 5

2

1

k

k k k k kk ( )( )( )( )( )

1 .01044926723267323166... 41

41

1

16 2 21

sinh

kk

.01049303297362745807... 16

7

8

1

56

2

2

2

8 8

3

8cot

loglogsin logsin

=1

64 8

1

822 2k k

k

kk

k

( )

...

0

3)4(

)1(

4

)3(3

216

197

k

k

k

=

1

0

1

0

1

0

333

1dzdydx

xyz

zyx

... 1

24576

1

8

3

8

5

8

7

83 3 3 3 ( ) ( ) ( ) ( )

=( )

( )

( )

( )

1

4 3

1

4 1

1

4

1

41

k k

k k k

.01065796470989861952... 2 5 2 33

24

5

43

9

82

81

16 ( ) ( ) ( ) ( ) ( ) ( )

= H

kk

k

14

1 2( )

.01077586137283670648... 13

8

2

4

3

12

3 1

3 1 8

3 1

3 1

log loglog

=1

12

1

12

1

12

1

121

1221 1k k

hgk

k

kk

k

( )( )

.01085551389327864275... log2131

192 6 52

dx

x x

6 .01086775003589217196... 1

256

1

4 13

3

40

1

( ) log

x dx

x

... 60

49

3600

1

2 1 2 2 2 6 2 70

( )

( )( )( )( )

k

k k k k k

... 9

43 5

17 5

2

( ) ( )k kk

= 1

1 22 5 15

1 1k k kk

k k( ) ( )( )

.01105544825889466389... 11

1536

1

4

3

4

1

2 33 3

40

( ) ( ) ( )

( )

k

k k

1 .0110826597832979188... 1

2 1 255

11 ( )

( )k k

kk k

k

1 .011119930921598195267... 1

22 2cosh cos

=

sin sin sin

( )sin

( )/ /1 11 2

2

1 2

21 4 3 4 i i

= 114 4

1

kk

... 3 12 2 9 31

12

130

log log( )( )k kk

.01121802464158449834... csc

( )cos cos sin sin

1

4 11 1 3 12

2 2

ee e

1

=1

14 41 kk

... ( )jkkj

3 115

... 1

36 6

1

8

23

0

( )k

kk

k k

k

... log log

( )3

2

2 2

36 r

... 32 2 8 2 4 3 2 4 51

2 6 51

log ( ) ( ) ( ) ( )

k kk

=( )k

kk

5

21

... 720 7 11

6

0

( )( )

x

e edxx x

... 720 7 17 ( ) ( )( )

...

3 2 3 1 1 1120

10

120

dx

x

x dx

x

... 21

71

/k

k k

...

sinh2

1

2 4

2

21

k

kk

=

1

02)1(

)log2cos(dx

x

x GR 3.883.1

... 117

2 7k ks ds

k slog( )

.01175544134736910981... coth

2

12e

...

1

125

1

5 12

2

50

1

( ) log x dx

x

.0118259564890464719... 2

1144

49

864 1 2 3 4

H

k k k k kk

k ( )( )( )( )

... 3 3

2 22 2 4

1

1 2

2 1

31

( )log

( )

( )( )

k

k

k

k k

... 3 2

6 4116

52

27

log x

x xdx

.0121188016321911204... 7

5760

5

192

93

256

1

1

4 2

2 42

( )

( )

k

k k

... e

k

k

k

2

0

7

32

2

5

( )!

.01227184630308512984...

256 162 20

dx

x( )

.012307165328788035744...

1

0

1

0

1

0222

333

132

)3(3

8

1dzdydx

zyx

zyx

.01232267147533101011... 8 1 .01232891528356646996... 5 2 6 3 ( ) ( )

.01236617586807707787... 4 2

2 41

30

768

1

2

1

4 1

( )kk

J373

.01243125524701590587... 18 13

32

9 2

32

13

128

G

log

J385

.0124515659374656125... 80

3

8

45320 2 48 3 560

1

2 1

2 4

4 41

log ( )( )k kk

.01250606673761113456... 1 4 4 8 6 3 12 52

3

61

( ) ( ) ( ) ( )( )

k

kk

.01256429785698989747... 160 44 215

24 80 2 20 3

1

2 13 41

( ) ( ) log ( )( )k kk

.01261355010396303533... 1

12 42 ( )kk

.012665147955292221430...

022 18

1dx

e

exx

x

1 .01284424247707... W3

.01285216513179572508...

( )log

66

1

k

kk

... 1

1536

3

4

1

4 33

40

( )

( )

kk

... 31 3

108

1

1 2 3

2

21

k k k kk ( )( )( )

... 3 3

44 2

24

13

4

1

1 2

2 1

3 21

( )log

( )

( ) ( )

k

k k k k

6 .01290685395076773064... 5

81 1

4 3

60

log x dx

x

... 209

225

1

2 5

1

21

Gk

k

k

( )

( )

1 .01295375333486096191... 1

23 1

21

2

1

16 4 10

log log log( )

ii

ii

kkk

... 110

2

... 1

216

4

3

5

62 2

2

6 31

( ) ( ) log

x dx

x x

1 .01321183515... ( )( )

1 12

1

k

k

k

k

... 8

1 .01330230301225999528...

0 2

12sin

2

1

2

1sin

1cos45

6

kk

k

... 7 3

4

7054

3375

1

8

7

22

2

8 61

( ) log( )

x

x xdx

... 1

43 3 12 2 2

1

12

1 3

30

( ) log( )

( )

k

k

k

k

=49

3600

1

2 1 2 2 2 6 2 70

( )( )( )( )k k k kk

...

2 4 2

4132

3

128

21

163 1

2 3

( )( )

k

kk

... ( ) ( )( )

2 6 465

8

1 4

1

2

42

k k

k kk

= ( ) ( )

1 23

1

k

k

k k 1

... cos sin( )!( )

1 11 1

4 2 10

e k kk

... 1

512

9

8

5

82 2

2

6 21

( ) ( ) log

x

x xdx

... iLi e Li e

k

ki i

k2 2 21

sin2

e

... e ie i /2

...

2

4 2coth ( ) ( ) 1

= ( ) ( )

1 2 4 111

16 4

2

k

k k

kk k

... dxxx

x

0

222

)3)(1(

log3log

12

3log

... 3 3

256

1

4 30

( ) ( )

( )

k

k k k

.01417763649649285569... 2 3

27

7

18 12 31

dx

x x( )

.0142857142857142857 =1

701

162

5

kk

2 .0143227335483157... F

kk

k !

1

5021 .01432933734541210554... 127

240 1

8 7

0

1

log x dx

x GR 4.266.1

3 .0143592174654142724... 1

1 Fkk

.01442468283791513142... 3 3

250

2

61

( ) log

x dx

x x

... e k

k

( )

1

1

1

1 .01460720903672859326... e EikH

kkk

k

12

1

2

1

2

2

21

21

log

!

.01462276787138248... ( )2

142

k

kk

1 .01467803160419205455...

4

4196

7 4

84

1

2 1

( )( )

( )kk

AS 23.2.20, J342

... 27

28

8

)3(7

=

1

0

1

0

1

0222

444

1dzdydx

zyx

zyx

... 1

1728

11

12

5

12 12 2

2

61

( ) ( ) log

x dx

x

... 2 2

12

2

1

k

k

k

k

k

ke

k

( )

!/

...

1

12

332log

4

331

8

27log

4

1

kkkH

... e

erf e x x dxx4

12

0

( ) sinh cosh

... 2 3

44

Berndt 8.17.16

... 17

16 48 180

3

4

1

2

2 4

41

( )

( )k kk

.01521987685245300483... ( ) ( ) ( ) ( )( )

4 3 6 3 5 71

3

71

k

kk

... 2 2

40648

2 1

6 3

( )

( )

k

kk

... log ( )

( )( )

2

8 16

1

8

1

4 2 4 4

1

1

k

k k k

...

2

3 21

5

43

1

1 2

( )( ) ( )k k kk

... j6 J311

...

( )36

3

1 3

6

3 3

2

1 3

6

3 3

2

i i i i

= ( ) ( )

1 3 3 111

16 3

2

k

k k

kk k

... 45 5

484 3 504 2 462

1

1

1

6 61

( )( ) log

( )

( )

k

k k k

2 .01564147795560999654... ( )8

... 1

61 ( !)kk

... 2 6 42 4 252 2 4621

16 61

( ) ( ) ( )( )

k kk

... 13

1 k kk

...

1

8192

1

4

1

16 42

30

( )

( )kk

1 .0158084056589008209... log

log log( )2

2

1

2

2 1

2

2 1

2

2

21

k

kk

... dx

x52 1

.01593429224574911842...

( )4 1

4 14 11

kk

k

1 .01594752966348281716... 104

98415

6

6

G J309

.01594968819427129848...

1

3456

1

3

243 32 3

124416

1

3456

5

32

32

( ) ( )

= 1

12 8

13 3

1728 2592 331

3

( )

( )

kk

.01600000000000000000 = 2

125

2

61

log x

xdx

1 .01609803521875068827... 1

41 k k

k

.01613463028439279292...

254

15log2log

xx

dx

.01614992628790568331... 92

32 3

1

1

2

2 32

( )( )

k

k kk

= ( ) ( )

1 31 2

1

k

k

k k 1

.0161516179237856869... 10163

6 4

2

1

ek

kk

( )!

.0162406578502357031... 31 5

8

7 3

32

2

48 2 1

2 4

41

( ) ( ) log

( )

H

kk

k

2 .01627911832363534251... 2

71

k

k k k!

1 .01635441905116282737... 1

51 k kk !

.0163897496007479593... 1 22 16

21

2 1

3

32

Gk k

k

k

log

( )

( )

... 2

6

2 2

2 2

1

2

sin sinh

= 11 4 26 2

2

1

1k kk

k

k

k

( ) ( ) 1

6

.0164149791532220206... 6 2 3 4 5 ( ) ( ) ( ) ( ) ( )

= 1 1

16 17 6

26

1 1k k k kk

k k k

( )

( )

.01643437172288507812... 7 3

512

1

8 4 31

( )

( )

kk

.0166317317921115726... ( ) ( )

13

2

k

k

k

k

.01684918394299926361... log

arctan7

6

1

2

3

6

3

3

5

3 5 22

dx

x x

.01686369315675441582... 11 5 16

2

1

1k kk

k

k

k

( ) ( ) 1

.01687709709059856862... 6155 63

1080

4 3

5 41

log x dx

x x

.01691147786855766268... 2

2112

29

36

1

1 2

3

k k k kk ( )( ) ( )

.0170474077354195802... 1945 6 1

66

( )

( )

... 13 3

216 324 3

1

432

2

3

1

6 2

32

31

( )

( )( )

kk

...

6

16

1 3

2

1 3

2coth cot

i i

12

2 2 32

2 3i icot

= 1

11 66

2

1

1kk

k

k

k

( ) ( ) 1

... 3 2 2

8

3

2 12 3

0

dx

ex /

... 6 22

4 221

2 32

log( )

k k

k

kk

k

...

6

9456 ( )

... 1

120 1

1

5

5

2

5

2

log

( ) !( )( )k

k km

k m

1 .01736033215192280239... 1

22 22 k kk

1 .01737041750471453179... sinhcosh cos

4

3 11

3 62

kk

... 18 2 16 2

2 1k kk

k k

( )

... sin1

...

180 , the number of radians in one degree

... tan1

... 17 1 16 1

2 1k kk

k k

( )

... ( )

log( )6 1

11

6

2

k

kk

k k

... ( ) ( )( )

3 3 433

16 3 41

k

kk

... 11

12 2 3

3

2

1

16 16

2 1

tanh ( )

kk

k k

... ( ) log( )5 1 1 1

1

5

2

k

k

k

kk k

... 6

2

3

2

1

2

2

6

62

sin

i

k

kk

.01765194199771948957... 2

21

8

144

1

12 3

G

kk ( )

.01772979515817195598... 1

6

11 2

18

3

4 1 21

log log

( )( )(

dx

e e e ex x x x 3)

... 1

144

1

12

5

12

7

12

11

121 1 1 1 ( ) ( ) ( ) ( )

=( )

( )

( )

( )

1

6 5

1

6 1

1

2

1

21

k k

k k k

... 15 1 16

2 1k kk

k k

( )

... sinh( )

32

3

2 1

3 3 2 1

0

!

e e

k

k

k

AS 4.5.62

... ( )

( )arctan log

1

2 2 1 93

1

3

10

91

1

1

k

kk k k

... ( )5 ... e4

.018355928317494465988...

16)1(

)1,6()4()3()5()2()7(3k

k

k

HMHS

... 7

82

4

14 2 16 2

2 1

( ) coth ( )k k

kk k

.01840776945462769476... 3

512 42 30

dx

x( )

... 7 3

16 192 2 1

4

41

( )

( )

k

kk

...

2

1)()(k

kk

... 5

33

1

2

1

4 2 1 2 31

arctanhk

k k k( )( )

... ( )

( )

3

64

1

4 31

kk

... loglog

23

2

1

8 5 32

dx

x x

.01891895796178806334... log log2

3

3

4

1

30 4 3

1

36 622

k kk

=( )k

kk

1

62

1 .01898180765636222260... ( )4 1

2

1

215

1

k

k kkk k

.01923291045055350857... 2 3

125

2

61

( ) log

x

x xdx

1 .01925082490926758147...

( )

( )

( )5

6 61

k

kk

Titchmarsh 1.2.13

3 .01925198919165474986... e d

e xx

2 2 4 1

20

( )

x

.01931537584252291596... 3

4

2

3

2

22 3

3

2

1

3

1

8

2 2

2loglog

log loglog

Li

= ( )

( )

1

2 4 22

k

kk k k

.01938167868472179014... 112

3 3

2

1

2

2 1

31

( ) ( )

( )

k

k

k

k

.01939671967440508500... G k

k

k

k2 16 128

1

2 1

3 1

31

2

( )

( )

.01942719099991587856... 2

5

7

8

1 2 1

2 42

( ) ( )!

( )!

k

kk

k

k

!

.01956335398266840592... J3 1( )

.01959332075386166857... 14

96

2

32

1

2

1

2 12

1 3

21

log( )

( )Li

k

k

k

kk

.01963249194967275299... 4

33

1

61 3

3 3

21 3

3 3

2

( ) i

ii

i

= 13 3 1

1

13 16 3

2 13

2k kk

kk k k

( ) ( )

=1

13

1

a non trivialegerpowerint

306 .01968478528145326274... 5

.01972588517509853131... 1

9 108

1

3 3

2 1

21

( )

( )

k

k k

.01982323371113819152...

4

4190

17

164 3

1

2

( , )( )kk

.019860385419958982063... 3 2

4

1

2

1

64 43 21

log

k kk


.0198626889385327809... 3

8

7

1920

4

5 31

dx

x x

.01990480570764553805... 3

64

3

32

3

41

31

2

1

log ( )k kk

k

k H

1 .0199234266990522067... log

log ( )5

2

1

5

1 5

25

r

1 .0199340668482264365...

2 2

2 22

2

16

5

8

1

12 2

k

kk k k

k k( )( ) ( )2

.02000000000000000000 = 1

50

... 1

22 1 30 k

kk

k

( )

.02005429455890484031... 2

9

7 2

241

1

1 141

42

log

log( )

logx

dx

x

x

x

dx

x

.02034431798198963504... 10

9

1

3

2 1

2 1

1

2 31

2

( )!!

( )!!

k

k kk

J385

.02040816326530612245... 1

49

1 .02046858442680849893... 62 5

63 61

( ) ( )

a k

kk

Titchmarsh 1.2.13

.02047498888852122466... 4 5 4 4 3 12

2

51

( ) ( ) ( )( )

k

kk

.020491954149337372830...

( )

( )

( )

( ( ) )3

3

6

1 13

14

k

k nk n not squarefree Berndt 6.30

1 .0206002693428741088... 1

9

8

9 190

csc

dx

x

.02074593652401438021... 7 3

8 322 2

1

64

1

4

32

2

6 21

( ) log( )

x

x xdx

2 .02074593652401438021... 7 3

8 32

1

64

1

4 1

32

2

40

1

( ) log( )

x

xdx

.020747041268399142... B

kk

kk

2

1 2 4( )!

1 .02078004443336310282... 13 3

27

2

81 3

1

54

1

3

1

3 2

32

31

( )

( )( )

kk

Berndt 7.12.3

33 .02078982774748560951... 416

3

3

2

2

0

( !)

( )!

k

k

k

k

.020833333333333333333 = 1

48

1 .020837513725616828...

2 3 11

21

1

2

1

2 5 31

( )k kk

= ( )2 3

21

kk

k

1 .02083953399112117969... 1

2

1

2

1

2

1

4 24 40

cos cosh( )!

k k

k

.020839548935264637084... 1

6 4 2 2 2

4 1

41

cot coth

( )kk

k

... ( )

!

k

kk

1

... 3

22

50125 5

1 25 1

csc cotlog

x

xdx

=

4 2 25 3 5

125 5 5

3

5 2

/

... 197

108

3 3

2

2

5 41

( ) log x dx

x x

... 1

8

1

23

1

2 2

2

20

1

, ,log

x

xdx

=1

2

1

2

1

23 3Li Li

...

9

1

18 6 3

9 3

6

1

18 6 3

9 3

6

i i i i

=1

3 1 31 ( )k 1k

... ( )k

kLi

k kk k

1 15

25

2

1

=691

3276011 ( )

... 1

47

... 1

3

2

3

3

3 1

1

31

21 1

( )

( )

( )

k

k k

kk

k

... ( )

logk

k kkk k

1

4

1

41

1

42 1 k

... 7

1 .02165124753198136641... 41

16 2 10

arctanh1

4

kk k( )

... 1

46

... 42 26 22

3

1

logk Hk

kk

...

k

k

kkk

2

)12(32

1

22

1arcsin22

0

... 7

30720

4 3

51

log x dx

x x

=1

45

... 2

218

3

2

29

16

1

1 3

( )

( )( )k k kk

...

2

54

2

9

1

9 3 3

5 2

2

1

9 3 3

5 2

2

i i i i

1

18 6 3

5 2

2

1

18 6 3

5 2

21 1i i i i

( ) ( )

= 1

11 3 1

3 22 1k

k kk k

( ) ( )

...

12 )32)(22(4

)2(

2

)3(

12

1

kk kk

k

... 4 211

4

1

1 2

1

1

log( )

( )( )(

3)

k

k

k

k k k

... log log

( )

2

3

5

24 1 40

1

x

xdx

=1

44

... 11

42 4

1 1

1 26 42

41

( ) ( )( ) (k k k k k )k k

... 17

16

3

22 4 4

0

4

log sin tan/

x x d

x

... 9

8 12

3 2

4

2

2

1

2

2 2

4 22

log log

( )kk k k

.02291843300216453709... 3

23

13

8

1

9

2 1

932 1

log( )

1

k k

k

kk

k

... 1

1542 kk

1 .0229259639348026338... 1

2 1 244

11 ( )

( )k k

kk k

k

... 50 43

128

28 2

128

71

1536

2 1

2

1

2 61 3

2G k

k kk k

log ( )!!

( )!!,

...

64

2 2 22

1

2 1 22

2 21

sin sec(( ) )kk

... )(,1

2log12

log

2sofzeroa

Edwards 3.8.4

... 31 5

8

7 3

32

7 3

4

2

48 48 42 2

2 4 4 ( ) ( ) ( ) loglog

=H

kk

k ( )2 1 41

... log k

kk2

2 1

.023148148148148148148 = 5

216

1

5

1 3

1

( )k

kk

k

1 .02316291876308017646... 2

2 21

2 3

6

1

12 1

1

12 11

( )

( ) ( )

k kk

=log x dx

x120 1

.02317871150152727739... 233

1920

2

32

1

1643

coth

kk

... 1

43

... 3 2

3148

2

4

1

4 2 1

2

log

( )kk k

k

k

= arcsin logx x

xdx

0

1

.02334259570291055312... 3 2 3 3 2 3 2 31

2

0

( ) ( ) ( )logx x

e edx

x x

.02336435879556467065... 5

36

2

6

log

=( )

( )( )( )( )

1

2 1 2 2 2 4 2 50

k

k k k k k K ex 107b

.02343750000000000000 = 3

128

1

3

3

1

3

51

( ) logk

kk

k x

xdx

... 2

2 4

2

21

sinh

k

kk

.02351246536656649216...

3

5

27

1

4

1

381

4

27

)3(26 )2(3

=

2

3173

2

3134

)3(6

1

381

8

3

)3(233

3 iLii

iLii

i

= dxxx

x

136

2log

... 2 328567

1200072

2

8 71

( )log( )

x

x xdx

=1

42 3 B

... 2

20

6 5112

115

144

1

5

( )

( )

k

k k

dx

x x

...

2

2 211

1

21

1

2 1

cosh( )kk

2

... ( )jkkj

2 114

... 1

162 6k k

s dxk slog

( ( ) )

... ( ) ( )

( ) /

1 2 1

21

1

21 20 1

3k

k kk

kk

k HW Thm. 357

... 16 2 4 2 2 3 41

2 5 41

log ( ) ( ) ( )

k kk

=( )k

kk

4

21

... 21

61

/k

k k

... G k

kk8 64

1

256

1

4

1

6144

1

4

2 1

4 1

22 3

2

40

( ) ( ) ( )

( )

=G

8 64 128

7 3

32

1

6144

1

4

2 33

( ) ( )

=1

41

... 1728

2035)3()5,3(

=

1

0

1

0

1

0

444

1dzdydx

xyz

zyx

3 .02440121749914375973... ( )

!

/k

k

e

kk

k

k

3

0

1

31

.02461433065757607149... 4 3 3

16

2

5 31

( ) log x dx

x x

1 .02464056431481555547... 2 1 272

64 2 12 2 22 32 1

24

23 2

1

( log ) log log ( )( )

H

k kk

k

.02475518879810900399... 61 6

72

1

1 2

2

21

3

k k k kk ( ) ( )( )

7 .02481473104072639316... 5

.0249410205141828796... log( )

5

6 6 62

k

kkk

.02500000000000000000 = 1

40

... ( ) ( ) ( )2 3 43

2

15 4

2

k kk

= ( ) ( )

1 41

1

k

k

k 1

... 1

4 2

=1000

3937 = meters/inch

.02553208190869141210... ( )3 1

8

1

8 113

2

k

kkk k

... 4430

21871

21

8

( )k

kk

k

=1

39

... 23

7

3

2

1 32

1

1

arcsinh

( )k k

kk

k

k

... 2

22 2

18 2 4 2

1

2

1

2 1csc cot

( )

kk

J840

... ( ) ( )

( )6 7 1

71

k

kk

HW Thm. 290

.02584363938422348153... k

k kx d

k

1

162 5

6

log( ) x

.0258653637084026479... 2 5 2 3 4 3 2 42 4

1

( ) ( ) ( ) ( ) ( ) ( )( )

H

kk

k

.02597027551574003216... 1

2 162 1

1

8 1 8 11

( )

( )( )k kk

GR 0.239.7

1 .02617215297703088887... 8

7

8 180

csc

dx

x

.026227956526019053046... 161

900 6 0

1

G

x x dx arccot5 log x

=1

38

.02632650867185557837... 1

2

7

21

2

12

2

0

e

k

kk

k

k

( )( )!

... 1233

2

1

ek

k

k

k

!

.02648051389327864275... log( )

22

3 1 41

5 42

dx

x x

dx

x x

... 3 1

3164 2

1

2 1

G k

k

k

k

( )

( )

... 3 3

4

7

8

1

3 30

( ) ( )

( )

k

k k

=

1

0

1

0

1

0

222

1dzdydx

xyz

zyx

...

1 2

)2(

2csc

2log

kk k

k

=

122

11log

k k

... 4 7 2 5 3 46

1

( ) ( ) ( ) ( ) ( )

H

kk

k

... ( , )

.02681802747491541739...

3

2

6

7

216

1

18

)3(13

381

52 )2()2(

3

=

2

3132

2

314

)3(3

1

2

)3(

381

102 33

3 iLii

iLii

i

=

125

2log

xx

dxx

... 1

81

1

3 13

3

30

1

( ) log

x dx

x

=1

37

... Gk

x dx

x x

k

k

8

9

1

2 5 20

6 41

( )

( )

log

... ( )

( )!cosh

k

k k kk k

1

21

1

2

1

2 2

... 2 4

4164 768

1

2 2 1

k k

kk

( )

( ) J349

... 3

2

5

324 2

0

4

sin tan/

x x dx

...

1 .0275046341122482764... B

kk

k

2

0 2 1( ) !

.02755619219153047054...

( )

( )

log

( )

log5

5 1

1

55 51

p

p

k

kp prime k

3 .0275963897427775429... cosh( )

!

e e

k

k

k2 20

GR 1.411.4

.02768166089965397924... 8

289 40

x x

edxx

sin

... ( )k

kLi

k kk k6

26

1

1 1

... 3

64 2

1

4 3

1

4 1

4 1

3

1

31

( )

( )

( )

( )

k k

k k k J326, J340

.02777777777777777777 = 1

36

1

2 1 2 2 2 4 2 50

( )( )( )(k k k k )k

K ex. 107c

= k

k k k kk ( )( )( )( )

1 2 30 4

... ( )

( )

( ) ( )

( )

1

4 4 1 4 42 1 1

k

kk k kk

k

k

k

k k

1

.02788022955309069406... 115 5

16

1

2 50

( ) ( )

( )

k

k k

... 7 183

2

0

ek

k kk

!( )

... 7 172

4

0

ek

kk

( )!

... 7e

... 1)2(sinh

cscsinlog2

11

1

kk

ke

e k

.02817320866780299106... 3 3

128 1

2

51

2

40

( ) log

x dx

x x

x dx

e x

... 1

...

8csc

8

3sinlog222log8

8

7cot

16

1

=

21log222log8

8

7cot

16

1

=

12

2 864

1

8

)(

kkk kk

k

.02839399521893587901...

145

21

2

cos

1cos

)1(cos1

51cos4

1cos21

kk

k GR 1.447.1

... 3

2

4 1

3

1

1

e

k

k

k

k

( )

( )!

=1

35

...

( )log

55

1

k

kk

... ( )ck

kLi

kkcc

kc

1 12

122

21

...

1

31

2)!1(

)1(

4

132

kk

k

k

k

e

... 3 3

22

4 1

2 3

41

( )log

log

( )

x

xdx

... log log11

30

12

x

x dx

= 122

34 2

1

6

7

6

2

3

1

72

2

3

7

61 1 2 2

log ( ) ( ) ( ) ( )

.0290373315797836370... 1

1000

9

10

2

5 12 2

2

51

( ) ( ) log

x dx

x

.0290888208665721596...

108 92 20

dx

x( )

3 .02916182001228824928... sinh

( )

3

7 31

3

2 21

kk

.02929178853562162658... 1

48

1

16

1 2

420

e k

k k

k

( )

( )!

1

... 4 1 2

2

k

k

k( ( ) )

... 1

34

.029524682084021688... c34 2 21

2412 2 8 3 3 2 6 4 ( ) ( ) ( ) ( )

Patterson ex. A4.2

.0295255064552159253... 7 3

4

56

272

1

2 1 32

( )

( )

kk

= log2

6 41

x

x xdx

= x x

xdx

4 2

20

1

1

log

GR 4.261.13

.02956409407168559657... 1

4

1

36

5

6

1

6

1

3 21 1

1

21

( ) ( ) ( )

( )

k

k k

= 1

36

5

6

4

31 1

6 31

( ) ( ) log

x dx

x x

...

0 2)12(22

1

8

1,

2

3,

2

3,1,1,

2

1

k k kk

kHypPFQ

...

12 )32)(12(4

)2(

8

)3(9

6

1

kk kk

k

1 .02967959373171800359...

3 2

40

1

1

4

163

41

/

log

x dx

x

.029700538379251529... 2

3

9

8

2 1

2 42

( )

( )!

k

k kk

!!

...

3 33

3 3 1330

log

x dx

e x GR 3.456.1

=

log x dx

x1 330

1

GR 4.244.2

.0299011002723396547...

2

214

39

16

1

1 2

k k kk ( )( )

J241, K ex. 111

... 3 2

2

2

2

5

4 2 1 2

2

1

log log

( )( )(

3)

H

k k kk

kk

.0300344524524566209... 2

27

3

21

2

1

log ( )

k kk

k

H k

2

... log ( )( )

6

5 51

52

kk

k

k

k

1 .0300784692787049755... ee erf k

k k kk

3

20 1

1

21

2

2 1 2 11

, ,( )!!

( )!! !(

)

2 .0300784692787049755... e erf k

k

k

k

1

2

4

2 10

!

( )!

.03019091763558444752... ( )31

2

1

21 1 i i

= )2()2(2

1)3( ii

=

1

1

235

1)32()1(1

k

k

k

kkk

... 3 2

4 401024

x dx

e ex x

.03030303030303030303 = 1

33

1 .03033448842250938749... 6 3

7

( )

... e

e

...

( )( )

3

42

2 2 c Berndt 9.27

1 .03055941076... j5 J314

.03060200404263650785... 1

64

1

4

1

16

1

4

1

2 16

1 2

1

cos sin( )

( )!

k

kk

k

k

2 .03068839422549073862...

2 2

322

1 2 35 5

1

2

( )( )

k k

k kk

= ( ) ( )( ( ) )

1 12

2

k 1k

k k

.030690333275592300473...

12

11

1

2

log ( )( )

kk

k

k

k

... dxee

xxx

022

34

18

3

240

.03095238095238095238 = 13

420

1

2 1 2 5 2 90

( )( )(k k k )k

.03105385374063061952... 132

1

2 1

3 1

31

( )

( )

k

k k

.03111190295912383842... 33

48 48

9 3

24

1

2

2

31

( ) ( )

( )

k

k k k

... 15 5

8

45 3

2140 2 126

1

1

1

5 51

( ) ( )log

( )

( )

k

k k k

... log

( )( )

( )(

2

4

1

61

1

1 22

k

k k kk )

5 .03122263757840562919... 2 2

2 1

1 22 1

1 1

k

k k

k

k k

k

( )

( )!sinh

... 2 1

216

2 2 31

1 2

log( )

( )( )

k

k

k

k k

... 1

32 84

1

16

1

21

csch

( )k

k k

.03125002980232238769... 1

2

5

32 151 1

k

kk

k

k

( )

.03125655699572841608... 23

2 2 81

2

22

3

log Li

1 .03137872644997944832... 1

51 ( !)kk

.03138298351276753177... 12635

3 9

1

1

2 4

5 51

k kk ( )

1 .03141309987957317616... cosh( )!

1

4

1

2 160

k kk

... 1

642

5

82

9

8

1

64

5

8

9

81 1 , , ( ) ( )

= ( )

( )

1

4 1

1

21

k

k k

= log x dx

x x6 21

... 12

2

1

11

12 2

2

logsinh

coth log

k kk

= ( ) ( )

11

2 11

k

k

k

kk

.032067910200377349495... 1

6 64

5 2

12 1 1

2

41

log log

( ) ( )

x

x xdx

... 527

7350

2 2

35

1

2 7

1

1

log ( )

( )( )

k

k k k k

=1

32

2

51

log x

xdx

... 123

2

29

6

1

2 2 6

1

0

log( )

( )

k

kk

k

k

... 1

3

=1

31

... )6(48

203)3(3)1,2,3( 2 MHS

... ( ) ( ) ( ) ( )5 3 3 2 4 5 2 15

=1

12 51 k kk ( )

... 5

2 4

1

1 2

2

2 21

k k kk ( ) ( )

... log

( )

2

4 2 2

2 1

8 2 102

G k

k kkk Berndt 9.6.13

1 .0326605627353725192...

15

155 )13(

1

)13(

11

243

)3(242

kk kkG

J309

1 .03268544015795488313... 1

31 k k

k

.03269207045110531343... 3

4

5

6

1 2 1

2 32

( ) ( )!

( )!

k

kk

k

k

!

... 96 646

0

dx

x

.03283982870224045242... 19

72

2

3

1

2 4

1

1

log ( )

( )( )

kk

k

H

k k

... 2

61300

log x dx

x x

.03302166401177456807... J

k kk

k

k1

0

4

41

4

1

( )!( )!

... 365

48

1

4 1

1

0

log( )

( )(

3)

k

kk

k

k k

... 7 3

16 192 2 1

4

41

( )

( )

k

kk

... 2

61

k

k k k!

...

3

3

2 6

1

61 3

3 3

21 3

3 3

2

2

( ) ( )i

ii

i

= ( ) ( )

1 3 2 111

15 2

2

k

k k

kk k

.0333333 = 1

30 2 4 B B

1 .03348486773424213825... 1

41 k kk !

... 3

30

.03365098983432689468... 1

27

1

3

13 3

81

2 3

7291

3

( ) ( )

= 1

1626

1

3

1

3 31 2

31

( ) ( )

( )

k

kk 1

.033749911073187769232...

1

02

6

)1(

)1log(2log3

34

5dx

x

x

.03375804113767116028... 5 2 3 4 51

1 51

( ) ( ) ( ) ( )( )k kk

= 1

6 52 k kk

.03388563588508097179... ( ) ( )k k

kk

0

1 2 1

1 .033885641474380667... pq k

kk

( )

21

.03397268964697463845... 3

2 12

2

2

2

2

1

2

2 2

3

log logLi

.03398554396910802706... ( )

( )!sinh

2 1 1

2 1

1 1

1 2

k

k kk k

k

.03399571980756843415... J Fk k

k

k4 0 1

0

21

245 1

1

4( ) , ,

( )

!( )!

LY 6.117

.03401821139589475518... 1

144

5

12

11

12 11 1

61

( ) ( ) log

x dx

x

.03407359027997265471... log

sin tan/2

2

5

164

0

4

x x d

x

1 .03421011232779908843... HypPFQk

k kkk

1

2 1 12 2

1

4

2 1

16 102, ,

, , ,( )

... 3 3

22 2 3

1

36 31

loglog

k kk

J376

... 1

2

11 1

12

1

ee

erf erfix

dx

sinh

... 4

2 34

2 2 11

3

4

2

20

1log( !)

( )!( )

k

k k

k

kk Berndt Ch. 9

... 7

6

7 170

csc

dx

x

... 1

29

... 1 1

3

1

3

1

1e k

k

k

( )

( )!

... 12 2 21

1 340

log( )(

)

k kk

.034629994934526885352... 3

2 122 2

3 3

4

12

4 32

log

( ) ( )k

k k k

... 1

1024

7

8

3

8

1

4 12 2

1

31

( ) ( ) ( )

( )

k

k k

... sin

arctan

x

xdx

0

1

2 .03474083500942906359...

22

3coscosh)(sinh

= sinhcosh cos cosh cos

4

1 1 3 1 1 33

= 11

61

kk

Berndt 4.15.1

1 .03476368168463671401... 1 11 1

1

2 12

2

2

e

k k e

k

kk

k

kk

k

k

/

/ ( )!

( )

=

Re( ) 2 1

1

k

ikk

2 .03476368168463671401... 1 11

11

1

2 11 2

e

k k e

k

kk

k

kk

k

k

/

/ ( )( )!

( )

.03492413042327437913... Ai ( )2

... 21

2

1

2 1 24 40

4

1

Lik

Hk

k

k

kk

( )

( )

...

1

2

1

4

1

2

1

2

1

2

1

2

i i i i

= ( ) ( )

1 4 1 111

15

2

k

k k

kk k

.035083836969886080624... 1

21 1

2

log( )

k

kkk

... ( ) ( )( ) ( )

( ) ( )( )

( )2 34

4

3 6

42 3

3

22 5

2

=H

k kk

k ( )

1 51

... 2 3 15

12 csc

AS 4.3.46, CGF D1

.03532486246595754316... 11

32

1

1

2

2 32

( )

( )

k

k k

... 1

9

...

3

4

1

2 2 1 2 2

1

1

( )

( )( )(

k

)k k k k J244, K ex. 109c

...

1

125

4

52 ( )

.03561265957073754087... ( )

( )

5 1

5

...

1

12

)12()1(

2

1

kk

k kii

=1

28

... 1

55

1p

k

kk

p prime k

( )log ( )

... 2

25 5

1 2 1 22

21

3

21

( ) ( )!

( !)

( ) ( )!

( !)

k

k

k

k

k k

k

k k

k

1 .03586852496020436212...

1

4)14()4()14()24(k

kkkkk

... 1

11 55

2

1

1kk

k

k

k

( ) ( ) 1

... 2 4 2

4132 384

7 3

16 2 1

( )

( )

k

kk

.03619529870877200907... 2 7 2

81

1

12 31

log

log( )x

dx

x

.03635239099919388379... 1

2

1

2

1

2 2 12 11

arctan( )

( )

k

kk k

... k

kk

k

k

k6

2

1

111 6 1

( ) ( ) 1

... 1

5

6

5

1

5 611

log( )

H nk

kkn

... 2 2 23

2

1

2

log

... log( )

log4

5 5 51

1

5

1

52 1

k

k kkk k k

1 .03662536367637941437... 1

36

1

6

1

6 5 11

21

60

1

( )

( )

log

k

x dx

xk

... 2 14

2 1

0e k

k k2

k

( )

!

...

1

1

27

2

1)27()1(1 k

k

k

kk

k

... 3 1

2 31128

3

32

1

2

1

4 1

( )

( )

k

k k

...

032 )4(256

3

x

dx

...

12 )22)(12(4

)2(

4

)3(7

4

1

kk kk

k

... dxxx

li log2sin1

2log225

1 1

0

... ( )5 ... 6

= 1

27

...

1235

1)38(1

kk

kkk

.037087842300593465162... 1

3

3

27 131 1

k

kk

k

k

( )

...

1225

1)27(1

kk

kkk

... ( )

log( )6 1 1

11

6

2

k

kk k

k k

2 .03718327157626029784...

21

3

5

32

...

03

)2(

)316(

1

16

3

8192

1

k k

... 9

4

3

2

7

8

1

2 122

log( )

( )

k

kk k k

... coth22 2

2 2

e e

e e

=1

2

16

2 20

k

kk k

B( )!

AS 4.5.67

.03736229369893631474... 1

2

3

64

1

4 1

2

2 31

( )kk

J373, K ex. 109d

... 1 2 4 32 4

1

( ) ( )( )

k

kk

...

13

)2(3

)912(

1

4

1

3456

1

432

)3(7

1728 k k

...

1215

1)16(1

kk

kkk

...

21

1!

1)5( 5

k

k

k

ek

k

... ( )

log( )5 1

11

5

2

k

kk

k k

... 3 2

4

19

20 8

1

4 4 11

1

42

1

2

log

( )( )

( )

k k

k

k

kk

k

.03756427822373732142... dxe

xdx

xx

xx

04

2

15

2

1

log

32

)3(

.03778748675487953855...

032 )3(348 x

dx

... 6

...

13

)2(

)69(

1

3

1

1458

1

k k

... 16 29

3

1

4 20e k

k

kk

( )

( )!

... 3 3

42 2

9

4

15 3

2

( )log

( )

k

k k k

=

13

1

)2()1(

)1(

k

k

kkk

.037901879626703127055... 1

2

2 2

3

1

27 331

log ( )k

k k k [Ramanujan] Berndt Ch. 2

...

12

51)5(

1

1

kk

kk

... ( ) log( )4 1 1 1

1

4

2

k

k

k

kk k

...

1

2)12(

812cos

7

1

k k

...

13

)2(

)58(

1

8

3

1024

1

k k

...

0

sinlog4

2log

22

1dxxxex x

1 .0381450173309993166... k

kk

5

52 1

.0381763098076249602... 80 4 48 2 6 31

2 12

3 31

log ( )( )k kk

.03835660243000683632... dxx

x

1

02)22(

)1log(

8

2log1 GR 4.201.14

1 .03837874511212489202...

0

1

1112

)1(

k

k

k

.0384615384615384615 =

0

5cos26

1dxex x

.0385018768656984607... cossin

cos sin cos22

22 1 1

=

1

21

2/32/1 )!2(

4)1()2()2(

k

kk

k

kJJ

31 .03852821473301966466... 3 3

.0386078324507664303...

022 )1)(1(2

1

2

1xex

dxx Andrews p. 87

= 1log4

log1

022 x

dx

x

x

GR 4.282.1

.03864407768699789287... dxxx

x

14

2

)1(

log

36

331

... dxxx 22/

0

23

sin488

... 7

8 162

21 3

2

64 20 8 1

2 2 1

2 4 3 2

42

log( )

( )

k k k

k kk

=k k

kk

3

2 2

( )

...

122

21

02

22

)1(

log

)1(

log)3(3)2(2

xx

dxx

x

dxxx

... 3

)3(

... 1log16

112log2

28

12

1

022

x

dx

x GR 4.282.9

.03886699365871711031... 21

2

1

2

5

41

1

2 2

1

2 11

arctan log( )

( )

1

k

kk k k

... 7 3

216

1

6 3 21

( )

( )

kk

=3929

100800

1

5 81

k k kk ( )( )

... 3

11

!

1)23(

22

1

k

kk

ekk

k

... 1

83 4 2 2 ( ) ( )i i

= 5

8 2

1

42 2

15

2

( ) ( )i i

k kk

= ( )4 1 11

kk

... ( ) log( )3 2 1 1

1

3

22

k

k

k

kk k

... 116

21

163

2 3

2

31

( )( )

k

kk

... 25 5

225 2

11

4

1

4 1

1

1

loglog

( )

( )(

2)

k

kk k k k

... 1

12 32 ( )kk

.03945177035019706261... 1

0

322

log)1log(54

12dxxxx

... 1)2(1

1

4

9log2

2

kk

k

k

1 .03952133779748813524... 2 1cos e ei i

3 .03963550927013314332... 30 5

22

( )

1 .03967414409134496043... 1cschcoth2

1 22

=

2222

1

1

1

212(2)1(

kk

k

kkkk

1 .03972077083991796413... 3 2

23

1

9 2 10

log

( )

arctanh1

3 kk k

= 1 21

64 43 21

k kk


...

02

)2(

)35(

1

5

3

250

1

k k

...

1

12

1)3()1(4

13

3 k

k kk

... dxx

xx

1

0

22

1

log)1(

4

5

3

.04000000000000000000 = 1

25

.0400198661225572484... 216

251)3()4,3(

=

1

0

1

0

1

0

333

1dzdydx

xyz

zyx

.04024715402277316314... ( )2 1

21

kk

k

1 .04034765040881319401...

3 1041 2 / ( )

4 .04050965431073002425... 7

.040536897271519737829...

15

2

)1()1,5(

2

)3(

4

)6(3

k

k

k

HMHS

.040634251634609178526... 13

30 8

2 1

161

( )kk

k

... sin cos( )

( )!

1

2

1

2

1

2

1

2 1 41

k

kk

k

k

... 23 1 5 1

161

2 1

4

1

sin cos( )

( )

!

k

k

k

k

... ( )( )

3 12 2

22

f k

kk

Titchmarsh 1.2.14

= 1

322 ( )jkkk

.04084802658776967803... 13

2

3

4

2

3

1

6

1

6

1

621 2

log log ( ) ( )

k k

k

k

k

kk

... 1681

292

1

2

383

2

1

2 4 2

1

1

cos sin( )

( )! ( )

k

kk k k

... ii eLieLi 2

12

1

...

54 2 8140

dx

x

.04132352701659985... B

kk

kk

2

1 2 2( )!

.04142383216636282681... 1

2

1

2 21 1

1

tanh ( ) k k

k

e J944

.04142682200263780962... 7 3

16 64

1

128

3

4

1

4 1

32

31

( )

( )( )

kk

1 .04145958644193544755...

3

1

3

12

2

3log

6 22

32

LiLi

.0414685293809035563... 11

3

2

3

3

2 3

1

3 1

1

1

e

e

k

k

k

cos( )

( )

!

.041518439084820200289... 112 4 2 2

12

4 22

tanh coth

( )k

k k k

.04156008886672620565... 26 3

27

4

81 32

1

27

4

3

32

2

5 21

( ) log( )

x

x xdx

2 .04156008886672620565... 26 3

27

4

81 3

1

27

1

3 1

32

2

30

1

( ) log( )

x

xdx

= x dx

e ex x

2

20

.04156927549039744949... 13

21

6 2 3

3 3

2

1

6 2 3

3 3

2

( )

i i i i

= 1

5 22 k kk

.04160256000655360000... 1

5

2

5 121

21

k

kk

k

k

( )

.04161706975489941139... 1

4 2

1

2

1

2

1

2

1

2

1

4

1

1

cosh sin cos sinh( )

( )!

k

k

k

k

.04164186716699298379... cos cosh( )

( )!

1

2

1

21

1

4

1

1

k

k k GR 1.413.2

2 .04166194600313495569... k

k Hkk !

1

.04166666666666666666 = 1

24

1

3 1 3 4 3 70

( )( )( )k k kk

C132

1 .04166666666666666666 = 25

24

1

221

k kk ( )

1 .04166942239863686003... 12

1 ( )!kk

1 .04169147034169174794... 1

21 1

1

2 4

1

4

1

40

cos coshcos

( )!

e

e kk

.04169422398598624210... ( )

! (

k

k kkj

jk )!

2 22

1

... 1

42

2

16

5 2 2

5 2 2

3

8arctan log

log

( )

arctan arctan2 1

8

2

81 2 1 2 4 4

2

dx

x x

.04171627610448811878... e

e

k

kk16

1

16

1

8 41

sin

( )!

1 .0418653550989098463... e

e kk3

2

3 3

3

2

1

3 10

cos( )

! J804

.04188573804681767961... 1

6

9

7 4 22

log

dx

x x

... ( )

log

15

2

k

k k k

721 .04189518147832777266... 3

4

6

...

1

22 2 1 2

1

2 1

log cos log sincosk

kk

GR 1.441.2

.0421357083215798130... 4 2 29

4

1

2 32

1

1

log log( )

( )(

)

k

k

k

H

k k

1 .0421906109874947232... 21

22

1

2 1 41 2 1 2

0

sinh ( )( )!

/ /

e ek k

k

= 11

4 2 21

kk

GR 1.431.2

.04221270263816624726...

1

2 4 2 2 2

4 1

24 4 41

cot coth

( )kk

k

1 .04221270263816624726... ( )4

2

1

2 114

1

k

kkk k

=

444 2

coth2

cot242

1

1 .042226738809353184449...

1)1(

122

22

12

2

1

12

1

kkkk

kkk

[Ramanujan] Berndt IV, p. 404

... 16 21

1 4

1

1 42

1

2

1

21

( )

( / )

( )

( / )

k k

k k k

.0422873220524388879... B k

kk

k

k

2 2

0 !

.04238710124041160909... log

( )( )( )(

2

4 24

1

4 3 4 1 4 1 4 31

)

k k k kk

J242

.04252787061758480159... 1 1

21

11

12

= 1

1

2 12 2

12 1

1k k

k

kk

k

( )

... 8

125

12

125

4

31

41

2

1

log ( )k kk

k

k H

.042704958078479727... 64

2 27 3

4

1

2 1

2

31

log

( )

( )k kk

... 1

2

1

2

1

2 2 11

cosh sinh( )!

4

k

k kk

... arctan log2

4 4

3

4 142

dx

x

.0428984325630382984...

2 2

31

102 2 3 4

( ) ( )

1 .042914821466744092886... 1

2

1

2 1 2csc

e dx

e

x

x Marsden Ex. 4.9

.04299728528618566720... 163

2

58

9

1

2 2

1 2

1

log( )

( )

k

kk

k

k

... dxx

x

1

06

3

1

)1(log

32

32log

2

32log53

.0431635415191573980... 1

6 12 3

3

4

1

3 2 3 1 3 3 11

log

( )( ) (k k k k )k

GR 0.238.3, J252

5 .04316564336002865131... e

.0432018813143121202... ( )k

kLi

k kk k

1 14

24

2

1

... 1

2

2 1

2 1

4

4

Mel1 4.10.7

.04321391826377225... i i ii i2 1 2 2 i ( ) cos( log ) sin( log )

.0432139182642977983... ( ) //

/ /( )2 1 1 4

2

1 4

11 4 3 42 1

1

2

e k

k

J114

.04321740560665400729... e k

k

2

1

.04324084828357017786... arctanh1 1

2 2

1

2 12 10e e k kk

log tanh( )( )

1 .04331579784170500416... ( !)

( )

k

k !k

3

0 3 1

1 .04338143704975653187... ( )3 1

2

1

214

1

k

k kkk k

.04341079156485192712... 23

4

13

22

5 2

2 2 1 2

2

1

3

loglog

( ( )(

kH

k k kk

k )k

.043441426526436153273... 2

3

2

21

1

2 2 1 2 31

arcsinhk

k k k( )( )

... ( ) ( ) ( )

1 4 4 111

15

2

k

k k

k kk

k k

= 1

23

... ( )k

kLi

k kkk k

1

2

2 22

22

2

.043604011980378986376...

13 )12(3)12(27

1

3

2log

4

3log

k kk


... arctan( )ee ex x 1

1

2 20

1

...

( )

( )

( )4

5 51

k

kk

Titchmarsh 1.2.13

.04382797038790149392... 3 2

4 8

1

12

1

4 4 1

1

42 2

log

( )

( )

k k

k

kk

k

1 .04386818141942574959... 16

15

1

4 2 1

2

0

arcsin( !)

( )!

4

k

k kk

... 23

6 2

8

33 2

0

4

sin tan/

x x d

x

... 7 2161

1 6

1

1 63

1

3

1

31

( )

( / )

( )

( / )

k k

k k k

... ( )

( )! (

1

1 2

1

1

k

k k k ) Titchmarsh 14.32.1

... 1

22

1

120

csch( )k

k k

... ee

ke

k

k

!0

... 1

16 2

1

4

21 2

1

( )k

kk

k k

k

...

245

245

1)4()3()2(2log

3

14

xx

dx

kkk

.04427782599695232811... 2 2 2 32

34 2

2 1

16 2 11

log log( )

k

k k kkk

... 1

4 48

1

2 2

2 1

21

5 31

( )

( )

k

k k

dx

x x

... 22

1 1( ) ( )k kk

.044444444444444444444 = 2

45 1 41

arccosh x

x( )

.0445206260429479365... ( )

( )

3

27

1

3 31

kk

... 3

32

1

4 12 31

dx

x( )

... 9 29 3

2

5

4

1

4 11

loglog

( )(

2)

kk k k k

...

1 5

1

5

1

kkLi

... 3

8 2

1

4

1

4 2 1

21 2

1

arcsinh ( )

( )

k

kk

k

k

k

k

... ( )

logk

k kkk k

1

2

11

1

21 1 k

... 13

... 144391

2

53

2 2 11

3

e

e k

k kk ( )!( )

1 .0451009147609597957... 2 21

8 2 10

arctanh1

2 2

kk k( )

... li( )2

... 8 27

2 22

1

log

Hkk

k

... 1 21

2

1

2

5

4

1

2 22 11

1

arctanh log( )k

k k k

=1217

26880

1

4 81

k k kk ( )( )

... ( ) ( )4 5

... 1

2 2

13 22

0

coth( )

/

kk

... 2 2

3

5

12

2 4

22

log ( )!

( )!

k

kk

=1

22

... 1

7

... 7 4

8

3 3

4

1

1

1

41

( ) ( ) ( )

( )

k

k

k

k

.0455584583201640717... 17

83 2

1 31

log( )

dx

e ex x

... 2

21216

2

36

1

6

( )

( )kk

... 32 2

1

16

1

16 22 1 log

=log

log

x

x

dx

x 2 20

1

216 1 GR 4.282.10

.04596902017805633200...

9 18 3

3

6

3 3

18 3

7 3

6

3 3

18 3

7 3

6

log i i i i

=1

27 131 kk

... 10

3

3

2

47

36

1

3 11

log( )(

3)

kk k k k

... log ( )( )

5

4 41

42

kk

k

k

k Dingle 3.37

= ( )( )

log

14

1

41

1

42 1

kk

k k

k

k k

k

... Ik kk

31

21

6

1

3( )

!( )!

1 .0460779964620999381... 1 22 2

1

2 4 21

( ) cot

k kk

=( )2 2

21

kk

k

.0461254914187515001... 8

2

2

1

4 2 4 1 41

log

( )( )k k kk

J251

= 1

4 1 211 ( )k j

kj 1

J1120

1 .046250624110635744578... F kkk

2 12

2 1

( )

.046413566780796640461... 5

6

4 2

3 6

3 3

4

5 3

4

14

2

logcot cot

( ) i i

k k

k

k

2 .04646050781571420028... 1

3 130

kk

.04655015894144553735... 1

2

3

12

1

6 1 6 11

( )(k k )k

.04659629166272193867...

2 1

1 2

1

2 21

2

1

4

1

2 2F

k

k

k

k

, , ,( ) ( !)

( )!

...

2 122

22 2

0

2

log logsin

/

x dx GR 4.224.7

= log2

20

1

1

x dx

x GR 4.261.9

.04667385288586553755... 11

41

2

1 2

0e

k

k k

k

k

( )

!( )

.04672871759112934131... 5

18

2

3

1

3 9

1

1

log ( )k

k k

... csc ( ) csc ( )/ /1 11 4 3 4

... H

kk

k

2

51 1( )

...

22

)(

kkk

k

1 .046854285756527732494... 1

1

1

2 22k k k k kkm

km( ) log log

... ( )

( )! !

k

k k kI

kk k

2

1 2

2 12

...

21 2

1

2

1

2 1

3

20

4

arctansin

cos

/ x

xdx

... sin

/

3

3

0

3

... 5 5log

.04719755119659774615... 3

12 1

2 4 2 1

2 1

16 2 11 1

( )!!

( )!! ( ) ( )

k

k k

k

k kkk k

k

... 3

1

21

1

6 5

1

6 11

1

arccos ( )k

k k k J328

= 2 1

16 2 10

k

k kkk

( )

= dx

x

x dx

x

x dx

x60

4

60

1 2

301 1

/

1

... 1

4 ( ) ( )i i

... 4

5193

30 4

31

( ) ( )a k

kk

Titchmarsh 1.2.13

93648 .04747608302097371669... 10 ... 2

...

2 4 4

413

7

909 3

1

( )log

( )

x

xdx

=1

21

... 5

43

1 1

3 25 32

31

( )( ) (k k k k k )k k

... log ( ) log( )

2 2 11

11 2

2 22 2

k k

k

kk k

1

... log

( )( ) (

2

4

1

8

1

4 2 4 4

1

2 112

2

)

k k kkk

k k

... 2 2

118

1

2 2

2

( !)

( )

k

k !k

...

1

5

1

21

21

2

1

4 32

i i

k kk

= ( )( )

12 1

41

1

kk

1

k

k

1 .0483657685257442981... 1

2 1 233

11 ( )

( )k k

kk k

k

... 1

36

2

3

7

6

1

3 41 1

20

( ) ( ) ( )

( )

k

k k

=log x dx

x x5 21

... ( ) ( ) ( ) ( )( )

3 3 5 3 4 61

3

61

k

kk

... 24 2 3 41

1081

22

0

1loglog( ) log

x x x dx

... 3 1 1

161

2 11

4

1

cos sin( )

( )

!

k

k

k

k

.04878973224511449673... 2 32035

864

2

6 51

( )log

x

x xdx

1 .04884368944513054097... ( ) ( ) tanh2 3

3

3

2

15 4

1

3

k k kk

222 .04891426023005682... 8

364 2

33 2

22

0

log ( )/x Li x dx

.04904787316741500727... ( ) ( )( )( )

2 2 3 41 2 3

1

k

k kk

= ( ) ( )

1 21

1

2k

k

k k 1

1 .04904787316741500727... ( ) ( )( )

2 2 3 39 11

1

2

2 32

4

k k

k kk

= ( ) ( )

1 11 2

3

k

k

k k

3 .04904787316741500727... ( ) ( )2 2 3 1 12

1

H

kk

k

4 .04904787316741500727... ( ) ( )( )

( )2 2 31

11 13

2

2

1

k

kk k

k k

.04908738521234051935... 64 4

2

4 20

x dx

x( )

5 .049208891519142449724... 49

16 2

4

0

e k

k kk

!

.0493061443340548457... log ( )3

2

1

2

2 1

3

1

27 32 11

31

k

k kkk k

= dx

x x4 22

.049428148719873179229...

41 1 1

1

11 4 1 4 3 4

42

( ) csc ( ) csc ( )( )/ / /

ik

k

k

... 1

87 6 3 3 27 3 12

1

32 2

log log ( )1

=H

k kk

k ( /

1 31 )

.04952247152318661102... 263

315 4

1

2 9

1

1

( )k

k k

.04966396370971660391... sech 3

2

13 3

e e

... 9

128 2

1

4 2 1

24

1

( )

( )

k

kk

k

k

k

k

...

1

1

3

1

3

1)1(

kk

k Li

.04974894456184419506...

1

02

4

)1(

)1log(2log2

42dx

x

x

... ( ) ( !)

( )!( )

1

2 1 2 1

1 2

1

k

k

k

k k

... 13e

... 1423 1

2 2

1 2

1

e

k

k

k

kk

( )

( )!

... 5 2

6

19

36

1

2 21

log

( )(

3)

kk k k k

... 115

2 5k kx dx

k xlog( )

... 2 1

2112

2 4 21

1 2

log( )

( ) (

k

)k

k

k k

.05000000000000000000 =1

201

92

4

kk

= k k

kk

( )

( )

6

3 21

J1061

1 .05007513580866397878... 2 31 2 1 4 1 2/ / / CFG F5

.05008570429831642856... ( )3

24

.05023803170814624818... 51

48 24

3

2

1

2

2

31

( )

( )k kk

.05024420891858541301... ( ) log

( ) ( )

( )2 2

3 3

4

1

1

1 2

31

k

k

k

k

2 .05030233543049796872... 12

log ( ) kk

.050375577025453... ( ) log

1 2

22

k

k

k

k

.05040224019441599976...

6

1

21 3

1 3

41 3

1 3

4

i

ii

i

= 1

2 1 31 ( )kk

1

.0504298428984897677... 8 22 4

3 2 12

4

0

1

G x

log log( ) log x dx

.05051422578989857135... ( ) log

( )

3 1

4 1

2

5 31

2

2 20

x

x xdx

x dx

e ex x

.05066059182116888572... 1

2 2

1 .05069508821695116493... 1

25

1

5

1

5 41

21

( )

( )

kk

= 2

125

1

16

1

25

9

5

21

( )

.05072856997918023824... Ik kk

40

21

4( )

!( )!

LY 6.114

2 .05095654798327076040... 2

21

1

2 21

sinh

kk

1 .05108978836728755075... cosh( )!

/ /1

2

1

2

1 1

20

e e

k kk

AS 4.5.63

2 .05120180057137778842... 21

51

/k

k k

.05138888888888888888 = 37

7201

12

113

logx

dx

x

.05140418958900707614... 2

51192

log x dx

x x

.0514373600236584036... 2

3 2

16 16

1

82 2

0

2loglog

x e x dxx GR 4.355.1

.05145657961424741951... 244

3

9

4

1

3 1

1

1

log( )

( )(

2)

k

kk k k k

1 .05146222423826721205... 22

5 5

3

5 csc

.05153205015748987317... ( ) ( ) ( )( )

2 6 3 6 47

8

4 1

1

2

42

k k

kk

= Berndt 5.8.5 ( ) ( )

1 11 3

1

k

k

k k 1

1 .0515363785357774114... H kk

k

( ) ( ( ) )3

1

1 1

... 504013700

17

1

1

e

k

k kk

k

( )!( )

2 .05165596774770009480... 1

2 2

2 2

24 2 121 1k k

k k

kk

k

( )

1 .05179979026464499973... 7 3

83

1

2 1 230

2

1

( )( )

( )

k

H

kk

kk

k

.05181848773650995348... 178

225

1

5

2 1

2

1

2 51

2

( )!!

( )!!

k

k kk

J385

.05208333333333333333 =5

96

1

2 4

1

1

( )

( )( )

k

k k k k

4 .05224025292035358157...

3 2

032

1

x x d

x

sin

cos

2 x

log log ( )3

4

2

3 4 3

1

36 6 621 2

k k

k

kk

k

.05225229129512139667...

dx

x x x x x x6 5 4 3 21

=

40

2

31

3

6

( )

( )

( )

k

kk

2 .05226141406559553132... Titchmarsh 1.7.10

ek

kk

cos sin(sin )sin

!3

1

32

.05226342542957716435...

( )

( )!( )

1

2 1 2 1

1

1

k

k

k

k k .05230604277964325414...

( ) !

( )!( )

1

2 1 2 1

1

1

k

k

k

k k .05238520628289183021...

1

1

1 1

3 33 31 1e e kk kk k

cosh( ) sinh( ) .05239569649125595197...

5 6 2

16 1 32

log

logx

x

dx

x .05256980729002050897...

k jk

k kk

jkkj

kk2 1 2 2 22

1 11

0

1( )

( )

k k

4 .0525776663698564658...

1

19 .052631578947368421 =

1

2

10

9

1

19

1

19 2 12 10

log( )

arctanhk

k k K148 .05268025782891315061...

.05274561989207116079... 8

1

6

2

4

1

4 2 4 31

log

( )(k k )k

3

3

12

2

3

2 1

41

log log ( )kk

k

.05290718550287165801...

.05296102778655728550... 2 24

3

1

4

2 1

432 1

log( )

1

k k

k

kk

k

17

720

1

2

4

40

( )

( )

k

k k .05296717050275408242...

3 2

2 2132 1

Gx

xdx

log

( ) .05298055208215036543...

1 .053029287545514884562... 2

22 2

164 8

1

8 1

1

8 7csc

( ) ( )

k kk

1

6 .05305164769729844526...

2 3 22

3

2 2

2 1 31

arcsin( )!!

( )!!

k

k

k

kk

6 .0530651531362397781...

3 3

2

7

42

1

1 32

( ) ( )

( )

k

k k .0530853547393914281...

log log2

4 31

2 2

0

1

1

x dx

x x

x x

x

dx

=

( )jkkj

113

1 ...

8 2 2 2 31

2

3

24 31 1

log ( ) ( )( )

k k

k

kk

k

1 .053252407623515317...

.05330017034343276914...

4

37

12

31

4

37

12

31

3

2log

6

iiii

=1

2 1 31 ( )k 1k

... 2 2 22

2

2 11

log log log csc( )

k

kkk

=

2 11

2 21

logkk

.053500905006153397... 1

40 ( )!k k!k

1

100

2

5

9

10 11 1

51

( ) ( ) log

x

xdx .05352816631052393921...

.05357600055545395029... ( )jkkj

5 111

.05358069953485240581... 1

31 ( )!k k

!k

( )k

kk

12

32

.05362304129893233991...

k

kk

!!

( ) !

40

.05362963066204526106...

1

11 Fkk

2 .05363698714066007036...

1 10

1

4

5

4

1

4

1

4 4 1F

k kkk

, ,! ( )

1 .0536825183947192278...

2

32

7 3

16

1

27 2 1

( )

( )

k

kk

.05391334289872537678...

43867

1436417 ( ) 3 .05395433027011974380 =

log log

( )

3

8

1

12 2 31

x

xdx .0539932027501803781...

3

41

3

2 31

logkHk

kk

1 .05409883108112328648...

2

2

11( )

( )k

k

k 1 .05418751850940907596...

.054275793650793650 =5471

100800

1

3 81

k k kk ( )( )

11

1

11e e

dx

e ex x

log( )

.054617753653219488...

33

2 21

12

7

6

2

31 1

log ( ) ( ) 6 .0546468656033535950...

log log1

13

0

1

xx dx =

2 2

111

2 1

2

( )

( )!!

k

k kk

!! 1 .05486710061482057896... J166

log ( )2

2

7

24

1

2 8

1

1

k

k k .05490692361330598804...

( ) ( )

( )5 6 1

61

k

kk

HW Thm. 290 1 .05491125747421709121...

51

5

1

5 1 52 20

2

1

Lik

Hk

k

k

kk

( )

( )

1 .05501887719852386306...

284

38

4 10

log( )(

3)

k

k kkk

.0550980286498659683...

log

log

2

4

1

4 12 20

1

2

x

dx

x GR 4.282.7 .05515890003816289835...

( )

!( )!

k

k kk I

k kk k

1

1

21

1

221

2

.0551766951906664843...

( )

( )!/2

11 1

1

2 1

1

2k

kk e

k

k

k

1 .05522743812929880804...

5

64 2

1 2

4

3

21

( ) ( )!

( !)

k

kk

k k

k .05524271728019902534...

2

31

6 3

48 2 2

( )

( )

k

kk

.05535964546107901888...

82 9 .0553851381374166266...

2

22

7

6

2

21 2 2 12

2

1

1

coth ( ) ( )

k

kk

k k

k

1 .05538953448087917898...

3

32 2 1

3 2

40

2

2 20

log logx

xdx

x

x xdx 2 .05544517187371713576...

x

xx dx

2

40

1

21

1

log GR 4.61.7 =

211

4 2

1

4 1

21 2

1

( )

( )

k

kk

k

k

k

k .0554563517369943079...

1

18 .05555555555555555555 =

1

22 2 2 2 2cos cosh sin sinh 2 1 .05563493972360084358...

4

4 2 10

k

k k k( )!( )

=

1

18

2

93

1

920

csch( )k

k k .05566823333919918611...

1

4

5

4

1

4 511

log( )

H nk

kkn

...

Hkk

k 2 11

2 .05583811130112521475...

1

42k

k k( )

0558656982586571151... .

38 .05594559842663329504... 14e

0

3/13/1

9)!2(

1

2

1

3

1cosh

kkk

ee 1 .05607186782993928953...

5

24

1

1

2 2

0

log( )

( )

x

x xdx 2 .05616758356028304559...

.05633047284042820302... k

k kx d

k

1

152 4

5

log( ) x

1 8 2 8 2 5 2 5 41

52 2

2 log log log log Li .05639880925494618472...

( )

( )

1

4 1

1

21

k

kk k

=

.05650534508283039223...

e

x x

xdx4 2

01 4

tan

GR 3.749.1

3

2

3

2

1

2

3

2

1

3

1

32

21

log log( )

LiH k

kk

kk

1 .05661186908589277541...

12 2 12

22

2

22

0

1

i i i i

Li e x x dxi log ( ) log( sin ) ...

3

4

4 73 .05681827550182792733...

222

22

)1(

1

4

3csch

4coth

4 k k

...

3

42

1

1 3

2 3

21 2

log( )

( )( )

( )

( )!

k !

k kk k

k

k .05685281944005469058...

1

2 2 1 2 21 k k kk ( )( )

= J249

= 1)2(1

1

kk

k

k

.05696447062846142723... 38 1

2

1 2

1

e

k

k

k

k

( )

( )!

.05699273625446670926... 1

8 8 2 2

1

142

csch sech

( )k

k k

.05712283631136784893... 1

22 3

14 3

2

( ) ( )k kk

= ( ) ( ) ( , )

1 3 11

1 1

k

k k

k d 2z k

.05719697698347550546... 7 675

120

4 3

4 31

log x

x x

.05724784963607618844... G x dx

e ex x16 4 40

1

21 1

1

2 1 2 10

Ei Eik kk

( ) ( )( )!(

1 .05725087537572851457... )

GR 8.432.1

= SinhIntegralx

xdx

x

dx

x( )

sinhsinh1

1

0

1

1

7 3

2

2

31

( ) ( )

r n

nk

5 .0572927945177198187...

( )log

( )3

42 2

7

720 124

14 2

5 42

k

k k k .0573369014109454687...

.05754027657865082526...

e

x

xdx4 21

4

1

cos

GR 3.749.2

14

2431

21

5

1

( )kk

k

k .0576131687242798354...

1

2 33

33 3

31

Li e Li ek

i ik

k

/ / cos

1 .05764667095646375314...

log log ( )4 2

0

12

4

1

1

x

xdx .05770877464577086297...

52 5 1 52

00 0 1

k

k kI F

( !)( ) ; ;

17 .05777785336906047201...

7 6

4

3

2

2

51

( ) ( )

H

kk

k

Berndt 9.9.5 1 .0578799592559688775...

log ( )

( )

2

2 2

2 1

2 2 1

1

2

1

22 11 1

k

k kkk k

arctanh .05796575782920622441... k

)

( ) (

1 11

1

k

k

k =

log log

( )

x dx

x1 21

= GR 4.325.3

( )( )

( )k

k

k

kLi

k kk k k5

25

25

1

5 11 1

1 .05800856534682915479...

.05807820472492238243... 3

2

7

6

2 1

2 32

( )

( )!

k

k kk

!!

...

11

21 2 2 1

1

21

1

2

arccoth i

ii

iarctan arctan

= sin

sin

/ 3

20

4

1

x

xdx

3 3

2 3

( )

.05815226940437519841...

1

1

( )

!

k

kk

2 .0581944359406266129...

.058365351918698456359...

12

21

2 )1(2

)1(

3

2

2

3log4

2

12

kk

k

k

kLi

2 10

11

4

5

4

1

4

1

4 4 1F

kkk

, , ,( )

1 .05843511582378211213...

.05856382356485817586...

( )2 1

2 11

kk

k

.05861382625420994135...

e

x x

xdx4 2

01 4

cot

GR 3.749.2

4

24 , volume of unit 4-sphere 4 .05871212641676821819...

.0588235294117647 = 1

171 2

44 1

0 0

( )cos( )k k

kx

x

edx

( )

( )

( ) log4

4 41

k k

kk

.05882694094725862925...

1

18

1

9

2

3

1

9 2

1

21

21 2 1

1

log arctan ( )k kk

k

kH .05886052973059857964...

1

2

9

8

1

17

1

17 2 12 10

log( )

arctanhk

k k .05889151782819172727...

H k

kk

k

sin

1

1 .058953464852310349...

log( )

log3

4 4 4

1

41

1

42 1

k

k kkk k

.05897703250591215633... k

3 .05898844426198225439...

3 5 46

3 6 21

2 1

1 1 122

12 3 4

( ) ( ) log

( )

H k

k k k kk

k

7

11520

1

2

4 1

41

4

( )

( )Re ( )

k

k kLi i .05918955184357786985...

8

1

3

1

2 1 2 1 2 31

( )( )(k k k .05936574836539082147...

)k

J240, J627

ee

erfkk

22

1

2

1

2

!!

1 .05940740534357614454...

10 !

!!

!!

1

2

1

2 kk k

k

kerf

ee

3 .05940740534357614454...

ee

erfk

kk

12

1

2 0

!!

4 .05940740534357614454...

.059568942236683030025...

1 12

11

1

2

( )( )k

kk

k

43

720

1

1 42

k k kk ( )( .0597222222222222222222 =

)

2

0

2

161 2 2 2( log log ) log

/

x x dx .05973244312050459885...

.05977030443557374251... 10 27 4

88 2

3 3

2

1

1

1

2 41

( )

( )log

( ) ( )

( )

k

k k k

1 .05997431887... j4 J311

= 50

3

... e erfk

k kk

12

2 11

( )!!

( )!! !

=e dx

x

x

10

1

... 1 12

1

20 1

4

21

e erfe k

k

k

k

, ,!

( )!

.06017283993600197841... 5

18

1

2 2

1

2

1

2 2 1

1 2

0

arctan( )

( )

k

kk

k

k

.06017573696145124717... 2123

35280

1

3 71

k k kk ( )( )

... 5 1 6 1

161

2

4

0

sin cos( )

( )

!

k

k

k

k

... 2 62

33

3

21

2 12 1

1

2

log log ( )( )(

k kk )k

H

k k

1 .06042024132807165438... ( )ck

kkc

1

12

.06053729682182486115... 112 9 24 2

1081

22 2

0

1

loglog( ) logx x x dx

.06055913414121058628... 3

512

5 .06058989694951351915... k k

kk

0

1 2 1

( )

... 3

2 21

2 1

2 91

( )

( )!!

k

k kk

!! CFG B4, J166

= 11

2 50

( )k

k k

.060930216216328763956... 23

2

3

4

1

3 11

log( )(

2)

kk k k k

... ( )( )

arctan

12 1 1

2 1

1 11

1 2

k

k k

k

k k

k

... sin( )

( )!2

1

2 11

1

4

1

2 2

k

kk k

.06147271494065079891... 4 2

3 6

7 3

4 6

7 3

41

logcot cot

i i

=( )

14

2

k

k k k

... 1

3

9 6 2 24 2 40

36 24 2 1

2

20

4

sin

( sin )

/ x

xdx

.06158863958792450009... 1

4

304

105

1

2 90

( )k

k k

... k

kk ( )2 12 21

.06160974642842427642...

3

4 4 22 23 4

1 4 1 4 /

/ /csc csch

= ( )

1

242

k

k k

.06163735046020626998... 9 81

24

1

2

1

2 4 1

1

1

cos sin( )

( )! ( )

k

kk k k

=5

811

51

4

1

( )kk

k

k

... ( )

( !)[{},{ , , }, ]

11 1 1 1

40

k

k kHypPFQ

... 35 40 2 6 31

1

1

4 41

log ( )( )

( )

k

k k k

.06196773353931867016... 2

25

3

51

2

61

1

2

( )k

kk

k

k

k

... arccsch4

.06210770748126123903... 216 1

3 2

4 21

2 2

20

1

log logx

x xdx

x x

xdx

8 .0622577482985496524... 65

... 1

16 4 22 2

1

8

1

21

csch

( )k

k k

... ( ) ( )k kk

1 22

1

= 1

16

1

11 2 1 15 2 2

2 1

k k

k kk k( )

( ) ( )

=x dx

x( )2 30 2

= log 11

4 91

x

dx

x

... 1

1 k kk

k

... 1

221

2k

k

.062515258789062500054... 1

241

k

k

1 .06269354038321393057... 2

2log

12

22

... ( ) ( ) ( )611

90

28

384 14 3 4 5

4 2

=1

16 41 k kk ( )

.062783030996353104594... 1

0

12 )1log( dxx

... 3

22

3 2

4

2 2

2 2 22 1

1

log logkH k

kk

... 5635

6 185 3 5

1

1

2 4

5 41

( ) ( )( )k kk

.06310625275909953582... log

log( ) ( )2 1

1

2

22

1 1

1

k

k

k

k

.063277280080096903577... 32 4

19

72 0

1

G

x x dx arccot3 log x

... 4 2

4 4145

10

335

1

1

k kk ( )

... 8 41

28

1

2

1

2 4 10

sinh cosh( )! ( )k kk

k

...

4

34

01536

1

1536

1

2

1

4 2

( )

( )kk

.0634363430819095293... 11

22

3

1

2 31 0

4

ek

k k k k kk k( )! !( )( )( )

... 8

3 32 3

12

2 1 20

Gk

kk

k

log

( )

Berndt Ch. 9

... 1

2 31

k kk

( )

Ik k

k0 2

0

1

2

1

16

( !)

...

= ek

kk

( )

( !)

!

1

22

0

arctan( )

( )

1

2

2

5

1

4 2 1

1

0

k

kk

k

k ...

... 1

5

... 1

4

4

4 141

4( )

log ( )

( )

logk

k

p

pk p prime

.063827327695777400598... 2 212

1

2

12

3 22

log( )

k

k k k

... 21

2

4

62

( )

( )

... ( )1 2

3

...

3

1)1(

... 12

1 k kk

... 4

3164

7 3

42

2 1

( )log

( )

H

kk

k

... ( ) ( )

1

2

1

1

k

kk

k

... ( ) ( )

( )

3 5

42

... 5

... 9

... 1

9 3

1 2

2

3

21

( ) ( )!

( !)

k

kk

k k

k

... 2

125

28 2

125 51

2

3

1

arccsch( )k

k

kk

k

... 205

34 2 3

1

1

2

3 41

( ) ( )( )k kk

... log ( )2

3

1

6

1

3 90

k

k k

= log 11

6 71

x

dx

x

... 1

486

2

3

1

3 23

40

( )

( )

kk

...

2 2

)()1(

kk

k k

.06451797439929041936... 1

1

1

11

2 12 2

12

1e

kkk

k

( )

( )

.06471696327987555495... 4

2 2 1 32

0

1

eEi

x x

edxx ( )

log

... 1

64

1

8

3

8

5

8

7

81 1 1 1 ( ) ( ) ( ) ( )

=1

32

1

8

3

8 81 1

2

( ) ( )

= ( )

( )

( )

( )

1

4 3

1

4 1

1

2

1

21

k k

k k k

.06487901723815465273...

1

02

3

)1(

)1log(

2

2log3

332

1dx

x

x

... ( ) ( )4 6

90 945

14 6 2

61

k

kk

... Ei

e

k

kk

k

( )( )

( )

( )

1 11 1

1

11 !

... log

( )2

41

4k

kk

... 11

12

3

18

3

2

1

9 31

1

322 2

log( )

( )

k k

k

k

kk

k

... log

loglog ( )2

2 3 3

2

4

3

5

2

2

2

3

1

2

2

3

1

2

1

3

3

16

Li Li Li

=log ( )

( )

2

0

1 1

4

x

x xdx

... 52 1

2

2 21

2

k

k kk ( ) ( )

1 ... 63

2

2 1

2

3 22

( )!!

( )!! (

k

k k kk )

... G

k

x

xdx

k

k2 8

1

2 1 121

2 21

( )

( )

log

( )

.06529212022186135839... 1

2

1

21

212

1

log sec ( )sin ( / )

k

k

k

k J523

.06530659712633423604... 10

61

2 21e

k

k

k

kk

( )

( )!

... ( ) log

1 2

2

k

k

k

k

.06544513657702433167... 241

144

8 2

3

2

2

1

5

2

1

log log ( )kk

k

H

k

...

02

)1()1(

)(

)1(

2

1

24

1

k

k

k

... ( )( )

13 1

3

1

3 11

13

2

kk

k k

k

k

.06557201756856993666... 819 3

2

81 3

23 3

4

32 1

log( ) ( )

= 1

3 13 21 k kk ( )

.06567965740448090483... 2 2

3

19

36

1

2 3

1

1

log ( )

( )(

)

k

k k k k

.06573748689154031248... 7 3

128

( )

.065774170245478381098... log( )( )csc ( )

1 1 1 22 2

1

k

kkk

1

.06579736267392905746... 2

21150

2

25

1

5

( )

( )kk

.0658223444747044061... 2

136

5

24 1 2 3

H

k k k kk

k ( )( )( )

... e

k

k

k

2

016

19

48

2

4

( )!

... log2

4 8

1

2 6 5 41

dx

x x x x3

... e e

.06604212644480172199... 31

31

2 1

2 3 2 11

arcsin( )!!

( )!! ( )

k

k kkk

... 2

2 2120 5 4 5 5

1

2

1

5 1csc cot

( )

kk

J840

... 2 3 4 4 2 101

12 41

( ) ( ) ( )( )

k kk

... csc( )cos ( ) sin

1

2 21 1 3 1

1

122 2

2 21

kk

=

12

2 1)2(

2

1cot

)1)(1(2

3

kk

k

.06633268955336798335... 61

256

3

4

1

43 3

3

4 21

( ) ( ) log x

x x

... log9

10

.06640625023283064365... 1

22 p

p prime

.06642150902189314371... 1

4

2

16 1

4 2

4 121 1

4 21

k

kk

kk

k

k k

( ) ( )

.06649658092772603273... 2

3

3

4

1 2 1

2 22

( ) ( )!

( )!

k

kk

k

k

!

... 1

1 eek

k

.06662631248095397825... 1 22

2

1

1

2 2

20

1

loglog log ( )

( )

x

xdx

.066666666666666666666 = 1

15

= k k

k kk

( )

( )( )

6

2 41

J1061

1 .066666666666666666666 = 16

153

1

2

1

4 3

21

1

202

2

,( )k

k kk

k

kk

.06676569631226131157... 1

2

8

7

1

15log arctanh

=1

15 2 12 10

kk k

( ) K148

.06678093906442190474... ( ) log3

18 1

2

41

2

30

x

x xdx

x dx

e x

... k k

kk

1

1 3

( )

... 1

21 k k

k

... 2 2 3 4 3 4 2 72 1

16 11

log log( )

k

k k kkk

... 2 2

3

2 2

34

1

23 3

2 3

3

log log( )

Li

=log

( )( )

2

120

1

1

x

x xdx

... 3

15

.067091633096215553429... csc cot

( )

( )2 2

2 2 21

1

4

1

4 1

2 1

k kk

k

... 12

ii eEieEii

=

1 !

sin,0,0

21

k

ii

kk

kee

i

... 48

2 2 113

20

12G

xx dx

log log log

=169

2700

1

3 61

k k kk ( )( )

... 1

2 10 5

1

25 1

2

2521 1

cot

( )

k

k

kk

k

... 90

13311

101

2

1

( )kk

k

k

... cosh ( )( )

31

2

9

23 3

0

!

e ek

k

k

AS 4.5.63

.06770937508392288924... ( ) ( )

( )

1 2 1

16

16

16 1

1

12 2

1

k

kk k

k k k

k

...

( ) ( )( )

( )2

11 1

1

1

2

1 1

k

kk

k

k kk k

...

e

e

x

xdx

2 2

2

12 20

1

cos( log )

... 2 1 2

0e

t

tdt

cos

AS 4.3.146

...

1

62 1

1

1 22

( )

( )

( )(

k

k kk )

Adamchick-Srivastava 2.22

.06804138174397716939... 1

6 6

1

8

21 2

0

( )k

kk

k k

k

... 1

4

1

1

2

2 21

log

e

e

dx

e ex x2

... log( )

( )( )2

5

8

1

1 3

1

14 3

2

k

k

k

H

k k

dx

x x

=dx

x x( )

1 31

1 .06821393681276108077... ek e

eke

e

1

0 !

... 122

1)(

kk

Hk

...

2

4 21 26 2

11 2

coth ( ) ( )k k

kk

k

k

1

... 4

4 12 2 20

1

dx

x x( log )( ) GR 4.282.3

... 7 3 2

72 41

2

log log x

xdx

... 919

13440

1

2 81

k k kk ( )( )

... ( )k

kk 22

... cosh( )!

/ /1

2

1

2

1 1

20e

e e

k e

e e

kk

...

2

31

2

31

9

)3(833

iLi

iLi

.06853308726322481314... log loglog( )3

22

1

2

1

4

1

52 20

1

Li Li

x

x

.0685333821022915429...

sin

3

2 31

32

2 kk

2 .0687365065558226299... 2

2 1 1 1 1 1 1 2 2 2 2 2 2 25

1

k

k k kHypPFQ

!{ , , , , , },{ , , , , , },

.06891126589612537985... log

( )k

kk4

1

4

1 .068959332115595113425...

05 15

4csc

555

2

5

2

x

dx

... 8

7291

1

3 1

1

3 1

4

4 41

( ) ( )k kk

... 22

71

2 1

2 81

( )

( )!!

k

k kk

!!

... 3

29

.0691955458920286028... 2 8

2

4 0

log log sinx x

edxx

.0693063727312726561... 12 2 3 14

542

0

1logarcsin log

x x x dx

... 1

512

7

8

3

8 12 2

2

41

( ) ( ) log

x

x

... ( )3 1

3

1

3 113

2

k

kkk k

... 1

1 1 1 1 2 2 2 2 13

1 k kHypPFQ

k !{ , , , },{ , , , ,},

.06942004590872447261...

32 2 2

2

2 30

x dx

x( )

.0694398829909612926...

3 9 36

184

3

1

3 1

21

2 21

log( )

( )

k kk

.06955957164158097295... log2

1

2

1

x

xdx

.06973319205204841124... 2 3

27

1

3 12 21

dx

x x( )

1 .06973319205204841124... 2 3

27

2

3 21

22 2

3

2

1

402 1

kk

k

Fk

, , ,

3 .06998012383946546544... 3

.07009307638669401196... 5

12

2

2

1

2 81

1

201

2

log ( )( )

( )k

k

kk

kk

k

= log( )

11

1 31

x

dx

x

.07015057996275895686... 7

9720

4 3

41

log x

x xdx

... ( )

( )!cosh

k

k kk k2

1 1

21

2 1

k

... e

e x dx

1

2 0

sin x

.0704887501485211468...

( )

log logcosh3 1

3

1

33

3

21

k

kk

1 .07061055633093042768... 41

4

1

4 13

42 20

2

1

Lik

Hk

k

k

kk

( )

( )

... 1e

e

e

e

.0706857962810410866... 4 2 3 41

5 42

( ) ( ) ( )k kk

= 1

1 41 k kk ( )

179 .0707322923047554885... 2 2

2 2 5

0

23 2

18 2 1

loglog log

( )(

x dx

x x ) GR 4.264.3

.07073553026306459368... 2

9

.0707963267948966192...

3

2 21 1

33 3

1

ie e

k

ki i

k

log logsin

1 .0707963267948966192...

1

2 1

sin k

kk

GR 1.441.1, GR 1.448.1

= arctansin

cos

sin1

1 1

2

21

k

kk

2 .0707963267948966192...

1

2

22 20

k

k k

k

... 1

1000

4

5

3

102 2

2

4 11

( ) ( ) log

x

x x

... 3

8

3

4

2

3

1

2 122

log( )

( )

k

kk k

... arctan log log sin2

5

5

20

2

5

2

0

x x

edxx

... ( ) ( )

1 212

kk

k

F k 1

3 .0710678118654752440... 5 2 48

2 12

0

k k

kkk

7 .0710678118654752440... 50 5 2

... 120326 1

60

e k k

k

k

( )

!( )

.071308742298512508874... ( )

log log log ( ) log3

48 2 2 2

66 12

22

0

1

x x dx

... iI e I e

k

ki i

k22 20 0

1

sin

( !)2

... ( )

14

1 2

1

k kk

k

kF

... 16

1

8

1

4 3 4 1 4 11

( )( )(k k k )k

J239

= log

log

x

x

dx

x 2 20

1

24 1 GR 4.282.8

.0714285714285714825 = 1

14

.0717737886118051393... 3 2

3 30432

x

e edxx x

... 1

3

1

2

1

21

1

31

1

332 k kk

.07179676972449082589... 7 4 31

16 1

2

1

kk k

k

k( )

.07189823963991674726... 1

4096

3

8

7

8 13 3

3

41

( ) ( ) log

x

x

... 10 3

82 2

0

4

sin tan/

x x dx

...

1 4

1

4

1

kkLi

=9

251

91

2

1

( )kk

k

k

... ( )kk

16

2

... 8

.07246703342411321824... 2

20

4 22

9

12

1

3

1

( )

( )

( )k

k

k

kk k k

= ( )cos

1 12

21

k

k

k

k

=log

( )

x

x xdx3

1 1

=x

e edxx x3 2

0

GR 3.411.11

.07246703342411321824... 2 2

0

13

12

1 1

( ) log( )x x

xdx

.072700105960963691568... 3

8 6 3

2 1

91

( )kk

k

... 1 21

2

1

4 2 1

1

1

arctan( )

( )

k

kk k

.07277708725845669284... 1

1000

3

10

4

53 3

3

4 11

( ) ( ) log

x

x x

... lim loglog

n k

n

nk

k

1

22

1

Berndt 8.17.2

...

11

1 1

1

12 2

2( ) k kk

=1 1

21 1

2

( )kk

k

.07296266134931404078... 12

5

4

13

4

2 1

4 4 1

1

1

k

k k

k

kk

( )

( )

...

0

2

64

12

4

12

kkk

kK

... cos2

10

x

edxx

.07325515198082261749... 1

20736

1

4

3

43 3

3

4 21

( ) ( ) log

x

x x

...

1

1

4 !

4)1(4

k

kk

k

k

e

... ( )( )

, , ,

11

2

1

81 2 3

1

222

21 1

kk

k k

k

k kF

k

=

16

9log1

2

1

... 120 2 2670 2 8520 2 8000 26 7 8 9I I I I( ) ( ) ( ) ( ) 3025 2 511 2 38 2 210 11 12 13I I I I( ) ( ) ( ) ( )

=k

k kk

4

21 5( !) ( )

... 7

4

...

2

2cosh

... 2

2 21

2 3

6

1

12 7

1

12 5

( )

( ) ( )

k kk

J346

1 .07351706431139209227...

1

2

)!2(

)2()!(

k k

kk

...

3 3

2

3

3

10

1

3 80

log ( )k

k k

...

12

1

)12(

)1(

k

kk

k

H

.07363107781851077903... 3

128

1

4 2 1 2 3 2 5

2

0

kk k k k

k

k( )( )( )

... ci( )

... 5

2

4 1

2 2

1

1

e k

k

kk

( )

( )!

.07388002973920462501... 35

48 2 CFG B5

.07394180231599241053... 8

3

11

3 24

0

4

sin tan/

x x d

x

.07399980675447242740...

2

2

2

21

2

1sinh

k

kk

1 .07399980675447242740... 12

21 1

sinh( ) ( )( ) ( )i i

... 3 62

33

2

3 3 12

1

log log( )(

H

k kk

kk 2)

.07402868138609661224... log log log

(log )( )2 3

2 3

2 3

2

2 2

32

1

2

1

2

5 3

8

Li Li

= log ( )

( )(

2

0

1 1

1 2

x

x x x )

.074074074074074074074 = 2

271

21

22 2

2

0

3

02 3

( ) ( ) ( ) ( )kk

k

kk

k

k kLi Li

= log log2

41

3

41

x

x

x

x

1 .0741508456720383647...

2

51

5

10

2( )

( )

( )

k

k k Titchmarsh 1.2.8

.07418593219600418181... ( )( )

11

1 13

23

2

k

k kkk Li

k k

2 .07422504479637891391...

155/45/35/25/1

11

)1()1()1()1(

1

k k

... 1

3

5

4 3 1 31

log( )(

dx

x x 2)

... 1

2

14

7

7

2

1

2 720

csch ( )k

k k

... 21

2

1

2 1 23 30

3

1

Lik

Hk

k

k

kk

( )

( )

=log2

21

2

02 2

x

x xdx

x

edxx

1

1 .07446819773248816683...

e ek e k

k

2 22

0

11

1log

( )

.07454093588033922743...

1

22

1

1

)1(

2

1csch

2

1

k

k

k

... Li2

4

5

...

2

12

04

0

1

16 2

1

16

1

4

1

4 1 1

G

k

x dx

xk

( )

( )

log

... ( ) ( ( ) )( )

1 4 11

11

14 4

2

k2

k k

k kk k

...

2

33

2

33

3

1

6

1

3

2 ii

= 2

3

1

6

1

31 1

11 3 2 34

2

( ) ( )/ /

k kk

= ( ) ( )

1 3 11

1

k

k

k 1

.074877594399054501993... 35 3

72

1

41

2

21

3

0

1

k kx x

k ( )log( ) log x dx

.07500000000000000000 = 3

40

1 .07504760349992023872... e

=5

66 5 B

2 .07510976937815191446... 1

40 ( !!)kk

.07512855644747464284... ( ) log( ) log3

16

1 4

0

1

x x

xdx

1 .07521916938666853671... 22

5

1

5( ) ( )

d k kk

Titchmarsh 1.2.2

.07522047803571148309... 1

4

1

2

1

2

1

2

1

2 1 2

1 2

1

cos cos( )

( )!

k

kk

k

k

1 .07548231525329211758... 5 3 3 2 ( ) ( )

... ( ) ( )( )

( ) ( ) ( )( )

2 34

42 3 2 5

1 41

H

k kk

k

... 14 3

22

1

27 3

1 3

31

log ( )k

k

k

k k Berndt 2.5.4

... 1

8

1

2

1 3

1e

k

k

k

kk

( )

!

.07591836734693877551... 93

1225

1

2 71

k k kk ( )( )

... 2031

2

1

4 21 4

1

1

ek k

k

kk

/ ( )

! ( )

.07606159707840983298... 190 4 1

44

( )

( )

... e

erfik

k

k k

k

1 11 4

2

1

1

( ) !

( )!

... e

erfi

kk

k

dx

e x

k k

kx1

1 42 1

1

1 0

1

( )

!

... ( )

log2 1 1

2 1

1

2

1

1

1

1 2

k

k

k

k kk k

= 12

2

1

2

log

arccoth kkk

.07646332214806367238... 1

41

21

2

3

2

3

2

i i i i

1 21

32

log( )k

k k k

... log ( )( )

237

607

1

70

bk

k

k

GR 8.375

= dx

x x8 71

... 12

1

2

1coth

2 2

e

J983

= 1

1

1

21 2 1 12

1

1

1kk i

k

k

k

( ) ( ) Im ( )

= sin x dx

ex

10

= dxx

x

1

0

1

1

logsin

...

2

1

2

1

2 12coth

e

= 1

1

3

21 2 12

0

1

1kk i

k

k

k

( ) ( ) Im ( )

.07668525568456209094... 1 5

25 2/

.07671320486001367265... 1 2

4 1

3

20

2

log sin log(sin )

sin

/ x x

xdx

GR 4.386.2

.07681755829903978052... 56

7291

81

2

1

( )kk

k

k

=1

13

... G G 2

... pk

kk

p prime k

4

1

4

( )

log ( )

... ( ) ( , )( )

( , , )39

83 3

1

31 3 0

30

kk

=1

2 1

2 2

0

1 x x

xdx

log

GR 4.261.12

=

1

0

1

0

1

0

222

1dzdydx

xyz

zyx

.07707533991362915215... 1

2

7

6

1

13

1

13 2 12 10

log( )

arctanhk

k k

... 2 2

40128

2 1

4 2

( )

( )

k

kk

=139

1800

1

3 51

k k kk ( )( )

.07720043958926015531... log3

41 1

x

xdx

.07721566490153286061...

1

2

1

120

1

log ( )x

x

xdx2 GR 4.283.4

= 21 12 2

0

x dx

x e x( )(

)

x

Andrews p. 87

... 1 10

1

( )x d

.07730880240793844225... 4

12604 T( )

... 9

162

3

2 12 1 1

2

2

log( )

( )k

kk

k

Adamchik-Srivastava 2.24

... 11

2

1

4 2 1

1

1

erfk k

k

kk

( )

! ( )

1 .07746324137541851817... 1

31 k kk !!

.07752071017393104727... log( cos ) ( )cos

2 1 121

1

k

k

k

k GR 1.441.4

.07755858036092111976... ( )k

k kkk k5

1

25 522

1

5 .07770625192980658253... 9 1 2 /

... ( )

( )arctan log

1

2 1 44

1

2

5

42

1

1

k

kk k k

3 .07775629309491924963... 0

2

1

0

112 2

( ) ( )k mk

n

kmn

mn

.07777777777777777777 = 90

7

.07784610394702787502...

cos( )

!e

e

k

k k

k

1 1

0

.07792440345582405854... 2 2 1 1 11 2 11

cos sin ( )( )( )

k

!k

k

k k

... 10514

4

p

p prime

Berndt 5.28

=

( )

( )

( )4

8 41

k

kk

Titchmarsh 1.2.7

= qq squarefree

4

.07795867267256012493... 8

92

2

3

1

2 1

1 2

0

log( )

( )

k

kk

k

k

... 50 43

128

1

7

1

2 7

2 1

21

2G

k

k

kk

( )!!

( )!! J385

... 1

log

... ( )

!/k

ke

kk

k

k

2

1

1

11

...

2

3

13 20e

x

xdx

cos

... 1

12 2 2

12 22 2

1

2 2 21

csch sech

sinh( )

( )

k

k k k

...

2

1

2

1

21

21

iiii

...

4 3

3

8

1

3 2 1

1

1

( )

( )

k

kk

k

k

...

sin sinh

cosh cos

2 2

2 2 2 21

= 1

11 44

2

1

1kk

k

k

k

( ) ( ) 1

=17819

226800

1

1 101

k k kk ( )( )

.078679152573139970497... 2

32

26 1

2 1

21

log ( )( )

k

kk

1

=

kk

kkk

32

2

11 1

2log

... ci x x dx( ) log cos log log sin1 2 1 2 20

1

... 121

kk

k

...

1 2

1)12()1(6

22log

3

2

k k

k

... 1

124 24 2 27

1

2 22 2

21

log( )k

k k

... 3 3

4 12

1

1

2 1

31

( ) ( )

( )

k

k

k

k

...

primep p 41

1log)4(log HW Sec. 17.7

... log ( )( )

4

3 31

32

kk

k

k

k Dingle 3.37

.07921666666666666666 = 7

961

12

19

logx

dx

x

... 2

)3(3

24

)6(53)2,1,3(

2MHS

... 1

2

1

22

1

2

1

2 1 21

cos sin( )

( )!

k

kk

k

k

... 3 2 21

1 21

log( )

( )(

)

k

k

k

k k

=1

16

2 1

1631 1k k

k

kk

k

( )

.07957747154594766788... 1

4

...

1

125

3

52 ( )

.07998284307169517701... 32 3

3 3

2

1

3

4

3

1

3 11

21

log

( )( )

k kk

= log( ) log1 3

0

1

x x dx

.080000000000000000000 = 2

25 10 11

( )kk

k

.080039732245114496725... 2 3251

10842

2

5 41

( ) ( )log( )

x

x x

= x x

xdx

3

0

1

1

log

2 .08008382305190411453... 3 92 3 3/

... 7

61

2 1

2 71

( )

( )!!

k

k kk

!!

.08017209138601023641... 4 3

411215

log x

x xdx

.0803395836283518706... log

( )(

3

2 6 3

1

6

1

3 2 3 31

k kk )

= ( )

( )

k

k k

dx

x x xkk k

1

3

1

3 3 12 26 5 4

1

.08035760321666974058... log log2 2 2

.08037423860937130786... 11 5 14 1

2

1

1k kk

k

k

k

( ) ( ) 1

... 1

3696 2 18 2 55 1

42 1

1

log log ( )

k k

k

H

k

... 1

1 113 2

( ) ( )k mk

n

kmn

mn

... log

( )

3

2 6 1

k

k kk

... 2 11

2cos e ei i

.08065155633076705272... 1

4

1

2 1 222

1

( ) ( )e

k B

kk

k !

.080836672802165433362... 10

50 10 5 51 2

25

0

1

x

xx dxarctan( )

... ( )

log

14

2

k

k k k

... 1 21

1

1

21

2

log ( )k k

kk

K ex. 125

= GR 6.441 log ( ) x dx1

2

=

1 1

1

1

20

1

log logx x

dx

x GR 4.283.2

122 .081167438133896765742... 8

63120 6 1

1

65

5

0

1

( )log( ) x

xdx

.08126850326921835705... 13

15 4

1

2 708 6

1

6

0

4

( )

tan/k

k k

dx

x xx dx

.08134370606024783679... k

kk

k

k

k

2

62

1

111 6 2

( ) ( ) 1

.08135442427619763926... 9 21

31

2

2 1 9 2 11

arcsin( )!!

( )!! ( )

k

k kkk

2 .08136898100560779787... 1

2 220log

x dx

x

.08150589160708211343... 1

22

3

4 11

loglog

( )(

dx

e e ex x x 2)

1 .08163898093058134474... 3

3 30

3 3

321

1

2 1

1

2 2

( )( )

( ) ( )k

k k k

.08183283282780650656... k

kk

k

k

k

3

72

1

111 7 3

( ) ( ) 1

1 .08194757986905618266... 4 1

2

1

16120

sin( )

!( )

k

kk k k

.08197662162246572967... 153

26

3 11

log( )(

2)

k

k kkk

.081976706869326424385...

1 )!2()1(2

3

2

1

1

1

k

k

k

B

e

e

e

1 .081976706869326424385... 1

2

1

2 2

1 2 1 2

1 2 1 2coth/ /

/ /

e e

e e

= B

kk

k

2

0 2( )!

AS 4.5.67

.082031250000000000000 = 21

2561

71

2

1

( )kk

k

k

.08217557586763947333... 1

2 6 3

2

6

1

6 809 3

1

log ( )k

k k

dx

x x

.08220097694658271483... log

log loglog2 2

2 2

2

22 3

3

2

1

3

1

2

Li Li

.0822405264650125059... 2 2

2212

2

2

1

2

1

2

logk

k k

1 .08232323371113819152... )1,1,2()4(90

4

MHS

1 .082392200292393968799... 4 2 2 11

4 60

( )k

k k

.0824593807002189801... 7

.08248290463863016366... 52

34

1

8

2 1 1

8 1

22

0 0

( ) ( )

( )

k

kk

k

kk

k k

k

k

k

k

k

.08257027796642312563...

1

16 8

2

8

2 2

2 2

cothsin sinh

cos cosh

= ( )8 4 11

14 4

2

kk kk k

.08264462809917355372... 10

121

1

10

1

1

( )k

kk

k

1 .08269110766346817971... H

k

ek

k !

1

6 .082762530298219680... 37

.08282126964669066634... ( )7 3 11

14 3

2

kk kk k

... ( )

log( )6 2 1

11

2 6

2

k

kk k

k k

.08285364400527561924... 328

)3(7

4

3

64

1 3)2(

= )()(28

)3(733 iLiiLi

i

= dxx

xdx

x

xx

14

21

04

22

1

log

1

log

...

4/14/14/14/14/1 2

1sinh

2

1cos

2

1sin

2

1cosh

216

1

4/14/1 2

1sinh

2

1sin

28

1

=

1

21

)!4(

2)1(

k

kk

k

k

15 .08305708802913518542... 41

11

12 1

1

7 6

1

14

1e

k k

k

k

kk

k

k

k

( )( )!

( )( )

( )! Berndt 2.9.7

122 .08307674455303501887... log

( )( )( )

5

2

5

21

k

k km

k m

.0831028751794280558... 5

2

5

6

1

5 12

1

1

log( )

( )

k

kk

k

H

k

.08320000000000000000 = 276

31251

4

4

1

( )kk

k

k

.08326043967407472341...

2 3

3

2

18

2

32

1

2

15

363

log log( )

Li

= log ( )

( )

2

0

1 1

3

x

x xdx

.083331464003213147... ( )k

kk4

2

.083333333333333333333 = 1

126 2 1

1

14 2

2

( )kk kk k

2 .083333333333333333333 = 25

124 2 1 14

1

H kk

k

( )

1 .08334160052948288729... k

kk

!!

( )!21

.08334985974162102786... k

kk

3

51 3

257

322 9 3 27 4 27 5

( )( ) ( ) ( ) ( )

1 .08334986772601479943... k

kk

!

( )!21

.08335608393974190903... log loglog( )5

32

2

3

1

3

1

42 20

1

Li Li

x

x

.08337091074632371852... ( )4 1 1

21

22

4

k

kLi

kk k

2 .08338294068338349588... cos cosh( )

1 1 21

40

!

kk

.08339856386374852153... 1 22

1

k

k

1 .08343255638471000236... 1

22 2

2

44 4

0

cos cosh( )!

k

k k

.08357849190965680633... 1

4 11k

k

( )

1 .08368031294113097595...

1

21 1 1

12

1 4 3 44

2 sin ( ) sin ( )/ /

kk

.08371538929982973031... 47 2

2

21 2

2

85

4 2 1 2

2 2

1

log log

( )( )(

3)

k H

k k kk

kk

57 .08391839763994994257... 21e 8103 .083927575384007709997... e9

1 .08395487733873059... H ( )/

33 2

.0840344058227809849... 11

2 1 21

Gk

k

k

( )

( )

= k k k

kkk k

( )

( )

2 1

16

16

16 112 2

1

= 1

8 21

3

41

kkk

k

, Adamchik (29)

= log x

x xdx4 2

1

.08403901165073792345...

2 21

21

21 2

1

log sin ( )( )!

k k

k

B

k k AS 4.3.71

.08427394416468169797... 8 32

1

4 3 4 2 4 1

2

21

( )( ) (k k k )k

J271

.08434693861468627076... ( )kk

2

2

1

.08439484441592982708... ( )5 1 11

14 1

2

kk kk k

.084403571029688132... ( )

!

4 11

4

21

k

kek

kk

... 2

1

2

1

2

1

2

1

4 21

HypPFQx

xdx, , ,

cos

.08444796251617794106... logsinh

( )log( )

4 4 11

1

4

2

k

kk

k k

.08452940066786559145... ( )3 1 1 1 1

21

2 32

k

k kLi

kk k

.084756139143774040366... 11

2

1

2

1

2

12

1

cot( )

( )!

kk

k

B

k

.0848049724711137773... e i ei i 2 24/ log log

...

4 2

24

415 42

2

26

1

2

21

43 2

log

log( ) logLi

=log ( )

( )

3

0

1 1

1

x

x xdx

.085000000000000000000 = 200

17

... 21

2 4 1

2

20

4

arctansin

cos

/

x

xdx

4 .0851130076928927352... k

dkk

2

... dx

x( )1

2

... 178

225

1

6

2 1

2

1

2 61

2

( )!!

( )!

k

k kk

J385

.085409916156959454015... 1

0

222

log)1log(2

)3(72log4

212 dxxx

.08541600000000000000 = 41

480

1

2 61

k k kk ( )( )

...

5

241

1

1

2 2

30

log( )

( )

x

x xdx

...

5

24

1

1

2 2

20

log( )

( )

x

x xdx

... 21

2

1

2 1 20

sinh( )!

k kk

= 11

2 2 21

kk

...

1 )14(4

1

kkk k

.0855209595315045441... 5

4

1

31

1

2

1

21

Eik k

k

k

( )( )

!( )

20 .08553692318766774093... e3

1 .08558178301061517156... 2

2 2 12

1

k

k kk ( )

Berndt Sec. 4.6

=

1

2

2

)(

kk

ku

... log k

k kk4 2

2

.08564884826472105334... log ( )2

3 3 3

3

4

1

3 708 5

1

k

k k

dx

x x

.08569886194979188645... log2

3

1

6

1 3

12

3 3

4

5 3

4

i i i

1 3

12

3 3

4

5 3

4

i i i

=( )

1

132

k

k k

.08580466432218341253... ( )

( )

1

22

k

kk k

1 .08586087978647216963...

3 2

2

3

4log GR 8.366.5

.08597257226217569949... 1

2

1

1 11

1

sinh

cosh cos

cos

k

ekk

.08612854633416616715...

3 1

31360

1

( )

sinh

k

k k k

1 .08616126963048755696... ee kk

k

12 2 1 2

1

2 1 10

cosh( )!( )

3 .08616126963048755696... ee kk

12 1 2

1

20

cosh( )!

.086213731989392372877... 1 1

8

1

2

1

22

12

1

sincsc

( )

( )!

kk

k

k B

k

... 2 22 2

0

2

48 8

2

4

2

4

log loglogsin sin

/

x x dx

.086266738334054414697... sech

1 .086405770512168056865... 7 3

41 2 2 2

64 4 2 1

4 2

31

( )( log ) log

( )

H

kk

k

.086413725487291025098... log cos sin

cos sin

/2

4 2 0

4

G x x

x xx dx

= GR 4.224.5 log cos/

x dx0

4

.086419753086419753 = 7

81 9 11

( )kk

k

.086643397569993163677... log 2

8

.086662976265709412933... 7

8 4

1

14 1

42 1

coth ( )k

kk k

.086668878030070078995... ( )

!

3 1 1 11

1 2

3k

k ke

k k

k

1 .086766477496790350386... 83

8 2 4 22 1

4 1

22

03

log log

( )

k

k kkk

.086805555555555555555 =25

288 1 2 3 4 5

3

1

k

k k k k kk ( )( )( )( )( )

.08683457121199725450... 1

1 2 30 k k k kk ( )!( )!( )!

.08685154725406980213... 1

8 32

2

2 1

2

31

log log

( ) ( )

x

x xdx

1

.086863488525199606126... ( ) log3 1 1 1

1

3

2

k

k

k

kk k

1 .08688690244223048609... log csc( )

1

2

3

2

3

2

3

2

2 1

1

k

k

k

k

.087011745431634613805...

642 2 1 2 2

0

2

( log ) log(sin ) sin cos/

x x x dx

... ek e

ek

kk

2

0

2/

!

... 2 1 2

4

1

4 2 1

21

1

arcsinh

( )

( )

k

kk

k

k

k

k

... 18

3 2

4

1

4

1

41

421 2

log( )

( )

k k

k

k

kk

k

=dx

x x x x5 4 3 21

... 30

3431

61

2

1

( )kk

k

k

... ( )311

96

6 8

6 85 4 31

k

k k kk

... 1

2

1

22

1

2 2 1 21

cosh sinh( )!

k

k kk

.087764352700090443963...

212 4

)4(

)14(4

1

14

1

kk

k

kkk

kk

...

12 )32(4

)2(

2

)3(3

18

1

3

log

kk kk

k

... logG

... 2 2 3 21 23

1

( ) ( )( ) (

)

k

k kk

... 2 2 33 1

2 1

1

23 21

24

( ) ( )

( )

( ) (

)

k

k k

k k

kk

k

... 2 2 71

2 2

12

2 2 11

arcsin

( )k

kk kkk

... 2465 1

50

4

21

e k k

x

xdx

k

k

e( )

!( )

log

... 6

2

3

2

31

16 4

1

log

logx

dx

x

1 .088000000000000000000 = 136

125

2 2

0

x x

edxx

sin

.08800306461253316452... J

k kk

k

k1

0

2 3

31

3

1

( )!( )!

.088011138960...

14

)(

k k

kpf

.08802954295211401722...

1

6

7

21

7

2 1 21

cos( )

kk

.08806818962515232728... 4 3

4 2116

6

log x

x xdx

6 .08806818962515232728... 4 3

40

3

20

1

16 1 1

log logx dx

x

x dx

x

=

0

3

xx ee

dxx

1 .08811621992853265180... 4

12 2 2

4 14

42 1

sinh

exp( )

k

ki i

k

kk k

.08825696421567695798... Y0 1( )

... 23

72 31

1

1

2

40

1

loglog( )

( )x

x

xdx GR 4.291.23

.088424302496173583342... where S is the set of all non-trivial integer powers

( ) 1

S

.0884395847555822353... ( ) ( )k kk

12

.08848338245436871429...

1

24

1)2,4(

jk jkMHS

... 1

25 1 3 1

1

2 2 3

1

1

cos sin( )

( )!( )

k

k

k

k k

.08856473461835375506... log tan ( )( )

( )!2

1

21

2 1

21

2 12

1

kk

k

k

B

k k AS 4.3.73

.08876648328794339088... 1

2 241

22

0

1

x x xlog( ) log dx

x

= x x x dlog log( )10

1

1 .08879304515180106525... log

( )

2

2

2 1

4 2 102

k

k kkk Berndt 9.16.3

= arcsin

logsin/x

xdx x dx

0

1

0

2

= x x

xdx

arcsin

1 20

1

=

log x

xdx

1 20

1

GR 4.241.7

= log1

1

2

20

1

x

x

dx

x GR 4.298.8

= log( GR 4.227.3 tan )/

20

2

x dx

= lo GR 4.227.15 g(tan cot )/

x x0

4

dx

= log(coth )coth

xdx

x0

GR 4.375.2

.088888888888888888888 = 45

4

... ( )

( )

k

k k k kkj j

jk

1

1

12 22

= ( ) ( )

1 121

k

mk

mk m

.08904125208589587299... 2 3

27 1

2

41

2

31

( ) log

x

x xdx

x dx

e x

= dxx

xx

1

03

22

1

log

.08904862254808623221... sin cos sin cosk k

k

k k

kkk

3 4

11

.0890738558907803451... 11 1

2

21e

erfe

xdx

x

( )

.089175637887570703178... ( )

( )

k

kk Li

kk k

1

1

112

22

2

1 .0892917406337479395...

1111 )!1(!)!2(

!)!12(2

4!1,2,

2

3(F

2

1

kkk kk

k

k

k

k

k

... ( )( )

12

1

2

1

222

21

kk

k k

k

kLi

k k

... 2

3

25

4

33

4

1 4

0

1

x dx

.08944271909999158786... 1

5 5

1

16

2

0

( )k

kk

k k

k

... dx

x k

dx

xe

k ( ) ( ) ( )0 0 0

1

1

... 2 2

7

319

2940

1

7

1

21

log ( )

k

k k k

... 1

3 16

3

8 16

7

8

2

4 8

2

4

3

8

cot cot logsin log sin

=( )

1

4 708 4

1

k

k k

dx

x x

... 10

38

2

3

1

2 20

log( )

( )

k

kk

k

k

... 90

180

1

2 441

4

( )

( )f k

kk

Titschmarsh 1.2.15

... ( )k

kLi

k kk k

1 13

23

2

1

... 4

... erfik kk

k

1

2

1

4 2 10

! ( )

=9

1001

91

( )k

kk

k

... ( )

( )log

k

k kkk k2 1

21

1

222 1

k

... 1

2

2

2

1

8

1

2

1

2

log i i i i

= sin

)

2

0 1

x

e edxx x

... ( ) log( )

2 2 21

2 22

3 31

Lik

kkk 1

... dxx

x

04

45

1

log

2512

57

... 2

11 5 5

15

... 5 5 1

2

1

5

2 1

0

kk

k

k

... 11 5 5

25

... 47

602

1

607 6

1

log( )k

k k

dx

x x

=13

144

1

3 41

k k kk ( )( )

... coth2

... 16 2 102 2

2

2

log( )

k

kkk

.09039575736110422609... 11

3

1

2

4

31

2 1

4 3 12 1

1

1

Fk

k k

k

kk

, , ,( )

( )

... 3 3

343 7

... 1

9

2

3

1

4

1

3 11

22

( )

( )

kk

= ( ) ( )k k

kk

1 1

32

... 2 41

2

4

3

1

4 2 11

arctanh log( )k

k k k

... csch( )kk

1

... 1

2 2

1

22

log

( )k

kkk

=

log 11

2

1

22 k kk

5 .090678729317165623... 1 10 1

1

22 4

1

1

2F ( , , )

( )!

k

k

k

c

kk

k

k !

... 11

111

ee

dx

e ex xlog( )( )

.090857747672948409442... e e

e

k

e

k

kk

( )

( )

( )

1

1

13

1 2

1

=1

11

=12

11 63

1

F kk

k

... 2

7

... 2

31

7 3

16 2 1

( )

( )

k

kk

... 1

2

6

5

1

11

1

11 2 12 10

log( )

arctanhk

k k K148

1 .091230258953122953094... 21

21

2

1

3 320

tan

k kk

... k kk kk k

( )( )

4 11

113 4

2

2

... 2

20

41108

1

3 3

( )

( )

logk

k k

x

x x

.09140914229522617680... 527

2 3

2

1

ek

kk

( )!

... 3 2

4 8

7

36

8

16

1

4

2

2

log ( ( ) )

G k kk

k

...

1 4

cos

1cos817

11cos4

kk

k

... 1

242 kk

... 1

G

... k k

k k k kk k

2

21

21

2

11

1

1

... 1

32 2

3 3

2

3 3

2

i i

= 13 1 14

2 1k kk

k k

( )

... 1

83 2

1

22

1

2 2 1 2

2

1

sinh cosh( )!

k

k kk

... 1

25

4

5

1

5 11

21

51

( )

( )

k

dx

xk 1

... 1

120

cos x

ex

=250

23

=3617

510 8 B

.09216017129775954758... log

log2

3

5

361

12

17

x

dx

x

.0922121261498864638... 245

64

5 4

4 21

log x

x x

... 4

2 11 2 10

log ( )( )(

k

)k

k

k k

=

122

1

02

2

)1(

11log

)1(

)1log(

x

dx

xdx

x

x

... log

( )(

3

4

3

36

1

3

1

3 1 3 31

)

k kk

= 5

541

51

2

1

( )kk

k

k

.09259647551900505426... ( )2

21

k kk

k

1 .092598427320056703295... ( )3 11

kk

...

1

1

)2)(12(

)1(

12

172log84

k

k

kkk

... ( ) log( )2 2 1 1

1

2

22

k

k

k

kk k

1 .09267147640207146772... 8

7

1

2 2 2 1 2

2

0

arcsin( !)

( )!

k

k kk

1 .09268740912914940258... 1

732 kk

... 61

32

33

2

3 21

Lix

x xdx

log

6 .093047859391469... k k

k

e k

k kk

k

k

2

2

1

21

1 1 ( )

!

( )/

.09309836571057463192... 1

2 10 20 5

10

10

cot

log

= 1

5

2

5 5

1

5

4

5

2

5cos log sin cos log sin

= ( )

1

5 708 3

1

k

k k

dx

x x

.0932390333047333804... 1 10

3

22 4

1 2 1F ( , , )

( )

!

k

k k

k

k

... ek k kk

21

8

1

2 40 !( )( )

... 18 13

32

1

5

2 1

2

1

2 51

G k

k kk

( )!!

( )! J385

.09334770577173107510... 22

8 12

13

1e

k

k kk

k

( )!( )

... pf k

kk

( )

31

.09345743518225868261... 5 6 2

9

1

2 1 2 31

log

( )(k k k )k

=3

321

33

8 11

2

12

1

( ) ( )( )k

kk

kk

kLi

k

... 1

232 5 1

1

12

log ( )( )

( )

k

k

k

k k

...

1

121

1 )!12(

)2(2)1(

2sin

1

k

kk

k k

k

kk

.094033185418749513878... 7

1920

1

4

1

16

4

42

42

44

44

Li e Li e Li e Li ei i i i( ) ( ( ) ( )

= ( )cos

1 14

41

k

k

k

k

... 9 3

4

2

2

1

6

2

4 2 321

( ) log ( )

( )

k

kkk

... 1 2 2 2 11 2

2 1

1

log log ( )( )(

k k

)k

H

k k

.09419755288906060813... 1

2

1

2

1

1

1

2 21

csc ( )k

k k

1 .09421980761323831942... 1 4

16

4 1

4 1 1

4 1

4 1

4

2

2

21

2

2

/ ( )

( )

( )

( )

k

k

k

kk

1 J1058

.09432397959183673469... 1479

15680

1

1 81

k k kk ( )( )

2 .09439510239319549231... 2

3

1

2

arccos

.09453489189183561802... 1

2

2

3

1

31

logk

kkk

J145

2 .09455148154232659148... real root of Wallis's equation, x x3 2 5 0

.09457763372636695976... ( )k k

kk

2

3 11

.094650320622476977272... Re ( )( ) ( )

ii i

2

2 .09471254726110129425... arccsch1

4

.094715265430648914224... 1

2

4 2 1

2

4 1

2 1 2 221

4

( )!!

( )! ( )( )

k

k

k

k kk

J393

.094993498849088801... 1

2 22

log k

kk

Berndt 9(27.15)

.09525289856156675428... 22

24 2 2 2 2

2 1

2 2 2 11

log log

( )!!

( )!! ( )

k

k k kkk

.095413366053212024... 8 21

2

1037

105

1

2 2 71

arctanh

kk k( )

1 .09541938988358739798...

1

2

3

4,

1 .095445115010332226914... 6

5

1

24

21

2 1

2 60 1

kk

kk

k

k

k

k

( )

( )!!

!!

... ( )

121

k

k k

k

... e Ei eEie

x

e xdxx( ) ( )

( )

2 11

1

2

0

1

... ( )1 1

3

... ( )1 4

3

.09569550305604923590...

( )3

2 11

kk

k

.09574501814355474193... 1

833 kk

.09587033987177004744... 5

3 2 41

tanh log xdx

x

... 2

2 2 4 1 2

0

1

G xlog log( ) log xdx

4 .09587469570927995876... 22

2 11

20

1

Gx

x dx

log log log

... 1

3

4

3

1

3 411

log( )

H nk

kkn

=12

1251

41

2

1

( )kk

k

k

.09601182917994165985... 5 5

54

4

41

( ) log

x

x xdx

... 1

64

3

8

7

8 11 1

41

( ) ( ) log

x

xdx

... 6

... ( )3

.09655115998944373447... 2 5 2 31 4

1

( ) ( ) ( )( )

H

kk

k

=

2

1

14

24

11)1,1,3()1,4(

k

k

jk jkjkMHSMHS

... log ( )

( )

2

2

1

4

1

4 1

1

1

k

k k k J368

= ( ) ( )

( )( )

1

2 6

1

2 2 2 40 0

k

k

k

kk k k

= 1

2 11k 2k k k k( )( )

GR 1.513.7

= dx

x xx x dx x d7 5

1

2

0

4

5

0

4

sin tan tan/ /

x

= log

( )

log

( )

x dx

x

x x d

x

1 131

30

1 x

= log 11

4 51

x

dx

x

= log

( log )(

x dx

x x 2 20

1

1 )2 GR 4.282.4

.0966149347325... ( ) log

1

11

k

k

k

k

... 2

0

2

09 2 1 2 1

2

2 2

k

k k

k

kkk k

!

( )( )!!

( !)

( )! J277

= J313 W2

= 1

6 5

1

6 11 k kk

J338

=log x dx

x60 1

... 2 3

3 3

2

3

2

32

2

32

1

2

log log

Li Li

=log

( )( )

2

12

320

1 x

x xdx

... e

e

k

ekk

2

121

( )

... ( )

( )

1

2 2 1

1

1

k

k kk

k

... e

k k

k

k

k

k

2

1 1

3

4

2

2

2

5

( )! !(k )

... e

k k

k

k

2

0

1

4

2

2

!( )

... kk

k

k

kk

k 4

1arcsin

)14(

144

22log3

2)1(

12

1

... 1

342 kk

.0974619609922506557... 1

37268

7

8

3

8 14 4

4

41

( ) ( ) log

x

x

... 2 5 2

8 2 2

1

4 2

1

21

sincsc sec

( )k

k k k

=22

225

1

2 51

k k kk ( )( )

... 9731

30

1

1 61

ek kk

( )!( )

...

iLi e Li e

k

ki i

k2

32

32

32

1

( ) ( )sin

... 312

3 21

4 2

2

3 21

log

k

k kk

= ( )( ) ( )

11

22

kk

k

k k

.0981747704246810387... 32 4

3 3

12 2

0

sin cos

( )

k k

k

dx

xk

1 .09817883607834832206... csc(log )

2ii i

.09827183642181316146...

log

5

4

.09837542259139526500... 166

3192

3

4 41

3

log( )

k

kkk

... 13 3

41 3

13

1

( )( )

( ) k

k k

=

1

0

1

0

1

0 1dzdydx

xyz

zyx

...

( )k

k kk4 2

2

... ( ) ( )2 2 1

2

2

1

k kk

k

... log( )

3 21

2

1

4 2 10

arctanhk

k k

= 11

27 331

k kk

[Ramanjuan] Berndt Ch. 2, 2.2

= Lik

k

kk

11

2

3

2

3

J 74

= sin

cos

x

xdx

20

1 .09863968993119046285... 31

3

1

3 12

32 20

2

1

Lik

Hk

k

k

kk

( )

( )

1 .098641964394156... Paris constant

.09864675579932628786... 1

2

2

42

0

1

si

x x d( )

log sin x

1 .098685805525187... Lengyel constant

.0987335167120566091... 2

2 2224

5

16

1

1

( )

( )

k

k k J401

.098765432098765432 = 8

811

81

4

41

( )logk

kk

k x

xdx

.098814506322626381787... 8 4

1

2

1

16 121

coth

kk

... ( ) ( )k kk

3 12

5 .09901951359278483003... 26

... log 2

7

... log

( )( )

( )

2

4

1

3 31

1

22

k

k kk

... ( )( ( ) )

11

2

2

2

kk

k

k

7 .0992851788909071140... k kk

kk

k

2

1

2

12 1 2

( )

... 9

... 7715

17282 5 3

5 31

( ) ( )( )

k

kk

... ( )( )

( )!

12 1

21

1

k 1

k

k

k

... 16

9

2

34 2

14

341

log

( )

( )!

k

k kk

!

... 24

51

4 12

1

log ( )( )

k kk

k

H

k

...

4

2

8

1

2 23

0

4

log

tan/Gx x dx GR 3.839.2

.0996203229953586595... 3

21

1

12

e Eik kk

( )( )!

... csc2

... 23

11

1 1

1 2

0

eEi

k

k k

k

k

( )( )

( )( )!

... e Ik

k

k

k

0

12

20

12

( )( )!

( !)

...

2

)(1log

kkk

k

.100000000000000000000 = 1

10

=

1 )3)(2(

)5(

k kk

kk J1061

.10000978265649614187... Likk

k10 10

1

1

10

1

1010 0

1

10

, ,

.10015243644321545885... | ( )|( ) k k

kk

1

91

.10017140859663285712... ( )3

12

.10031730167435392116... 3

2 12

1 1

2

11

12

32

2

( )

logk

k k kk k k

.1004753... 1

2

1 a non trivial

egerpower

int

...

1

0

1

0

1

0222

2222

14

16

3dzdydx

zyx

zyx

... 3

10

.100637753119871153799... Likk

k4 4

1

1

10

1

104 0

1

10

, ,

... 8 2

3

4 0

1

G

x x xarctan log dx

... 3 2 2

16 1 30

1

log log

( )

x

xdx

... 2 2

3

13

36

1

1 3

1

1

log ( )

( )(

)

k

k k k k

.1010205144336438036... 5 2 62 1

2 3 11

( )!!

( )! ( )

k

k kkk

1 .1010205144336438036... 6 2 61

12 1

2

0

kk k

k

k( )

... ( )3 1

2

.10109738718799412444... 1 2 2 2 2 6 2 6 13 2 3

0

1

log log log log ( ) x dx

... k kk k

kk k

2

1

4 4

4 32

4 11

1 ( )

( )

( )

... K3 1( )

... Likk

k3 3

1

1

10

1

10

1

103 0

, ,

... 2

)2(

2

log)1( 2

22

k

k

k

k

... 2

... ( )

( )log

1

2 1 3

3

3

4

32

1

1

k

kk k k

... 1

11 11

3

1131

1

1k

k

k

kk

k

( )( )

... 1

10

2 5

55

1

520

csch ( )k

k k

... log ( ) 22

kk

... k

kk !

1

.10177395490857989056... G

k kk9

1

12 9

1

12 32 21

( ) ( )

= x dx

e ex x3 30

.10187654235021826540... 1

2

1

3

1

2 2 1

1

0

arctan( )

( )

1

2

k

kk

k

k

... 8

9

20

27

4

3 4

2

1

logk Hk

kk

... ( )k

kk 21

.102040816326530612245... =5

49 7 11

( )kk

k

... e e

e

e

k k

e

e

k k

k

1 1

220

( )

!( )

.1021331870465286303... 75161

3 2

3 3

0

e k

k kk ( )!3

.10218644100153632824... 28

2

14

5

84

1

2 1 2 80

log ( )

( )(

k

)k k k

.10228427314668489686... 1 2

3

1

3 607 4

1

log ( )k

k k

dx

x x

1 .10228427314668489686... 4 2

33

0

2

logsin( ) log tan

/

x x

dx

... 1

2

2 1 2

21

2

0

1

cos log sin log sin

e

x x

xdx

x x

edx

e

x

... 1

101 ( )k kk

.10261779109939113111... Likk

k2 2

1

1

10

1

10

1

102 0

, ,

=37

360

1

6

1

21

( )k

k k k

= x x dxx

dx

x5

0

1

171 1

1log( ) log

.10280837917801415228... 2

2196

2

16

1

4

( )

( )kk

1 .10317951782010706548... ( )4 1

2 214

1

k k

kkk k 1

1 .10322736392868822057... 22

11113

22 2

1

231

k

k k

kk

HypPFQ

, , , , , , ,

2 .10329797038480241946... 5

55 3 13

2 1kk

k

k

k

( )

.10330865572984108688... 106

158 2

1

2

1

2 2 70

arctan( )

( )

k

kk k

1 .10358810559827853091... 1 21

log ( ) kk

.10359958052928999945...

( )

( )log3

42 2 3 2

2

4 21

2

31

x

x xdx

x dx

e ex x

= x x

xdx

2 2

20

1

1

log

GR 4.262.13

2 .10359958052928999945... 7 3

42 3 2

1

2 1 30

( )( )

( )

kk

= log2

20

1

1

x

xdx

GR 4.261.13

= x dx

e ex x

2

1

.10363832351432696479... 3

1 11

11

12

1

5

0e

k

k

k

kk

k

k

k

( )( )!

( )( )!

1 .10363832351432696479... 3

12 1

1

2

1e

k

kk

k

( )( )

( )!

.10365376431822690417...

2 1 20e

ex

xdxe

cos

AS 4.3.146

.10367765131532296838... 10

9

1

4

2 1

2

1

2 4

2

1

( )!!

( )!!

k

k kk

.10370336342207529206...

2

2

22

2

2 8

22

2loglog

log

4

= x e x dxx2 2

0

2

log

.10387370626474633199... 11

24

2 2 2

16

1

442

cot coth

kk

=

1

1 1)4(4k

k k

.1039321458392100935... 5 8 2 2 2

41

3

21

1

log log( )k k

k

H

k

... G k

k

k

k4

1

8

1

4 12 21

( )

( )

.10403647111890759328... dx

x x6 31 1

.10413006886308670891... 3

64 2 22 30

dx

x( )

1 .10421860013157296289... 1

3 3 10k

k k( )

... 2

21

1

3242 1

1

4 2 11

2

log( )

( )( )

k k

k

k

kk

k

... log !10

... 2

2 2

2

21sinh

k

kk

... 2 8

2

4

3

4

2

0

log log sinx x x

edxx

...

3 3

1

2

1

2 215 4

1

k k

k

dx

x x xk

! ( )!

( )! 3

.1046120855592865083... si x x x dx( ) sin log sin1 10

1

... 2

1

22

1

41

2 1 11 1

12

12

1

Fk k

k kk

, ,( )!( )!

( )!(

)!

... 8

9

9

81

81

1

log ( )

k

k

kk

H

1 .1047379393598043171... 4

1 11

2 21

erf erfix

dx

x

cosh

= 3

14

1cosh2

x

dx

x

... eee

... ( ) ( ) )( )

( )

1 3 14 1

11

1

33 6 3

3 42

k

k k

k kk k k

k

... 617

5880

1

1 71

k k kk ( )( )

... 71 431 2 3

2

1

ek

k kkk ( )! ( )

... ( )

( )

k

kk Li

kk k

1

113

23

1

... log( )

log

2 2

2

161

1

1612

1

k

k kkk k

=

1

1 10

1 2

2

x

x

x

x

dx

xlog GR 4.267.5

... 11

121

k kk ( )

... 7

42

1 1

14 22

22

( )( )k k k kk k

= ( ) ( ) ( )2 1 1 22

1

1

k k kk

k

k

1

=

rpoweintegertrivialnonaω

2 1

1

... 13

8

...

( )

!

k

kk

1

2

21

... 5

8

3 2

4

1

2 122

log

( )kk k

... 10 ... e1 10/

... sin ( )

( )

1

8

1

2 1

1 3

1

!

k

k

k

k

.1052150134619642098...

ci x x dx

2 0

2

sin log/

=2

19

... log log , ,10 91

10

1

101 0

1

kk k

= 21

19

1

19 2 12 10

arctanh

kk k( )

1 .1053671693677152639... 1

2 1 222

11 ( )

( )k k

kk k

k

... ( ) ( ) ( ) ( )( )

( )3 2 3 2 5

2

1 41

k k H

kk

k

.1055728090000841214... 12

5

1 2 1

2 4

1

1

( ) ( )!

( )!

k

kk

k

k

!

... ( )3 1

2 213

2

k k

kk 1k k

.10569608377461678485...

0

3 log4

1 2

dxxex x

... ii i i

k i k ik61 1

1 13 34 4

1

( ) ( )( ) ( )( ) ( )

.10590811817894415833... 2 3

32 2 3 2

2 2

332 2log

log log loglog

log

= ( )

( )

1

2 1

1

1

kk

kk

kH

k

.10592205557311457100... 4 28

3

1

2 2 61

log( )

kk k

1 .10602621952710291551... 4

41

1

16 21

sinh

kk

.10610329539459689051... 1

3

.10612203480430171906... 10

2

5

26

751

12

16

log

logx

dx

x

... 114

2 4k ks d

k log( )

s

...

4 42 20e

x

xdx

cos

= GR 3.716.4 cos( tan ) sin/

2 2

0

2

x x d

x

... k k

k

2 11

2

( )

.10634708332087194237... log log2

2

2

2

2

=( )

1

11

kk

k

kH

k

.10640018741773440887... loglog( )2

2

20

1

212

1

2

1

3

Li

x

x

.10642015279284592346... 1

82 2 2 21 1 2 2i i i i i ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) i

= ( ) ( )( )

( )

1 2 1 11

11 2

1

2

2 32

k

k k

k kk k

k

... 24 5 14 32

52 1

1 11 11

1

4 2

52

( ) ( ) ( )( )

k k k

k

3

k

= ( ) ( )

1 14

1

k

k

k k 1

=8

754

0

1

x xarccos dx

... 1

6

3

12

3

2

1

4 3

1

21

csch

( )k

k k

... ( )( )

arctan

11

2 1

11

1

32 1

k

k k

k

kk

k k

... 1

10

10

20

5

2

1

2 520

csch( )k

k k

... 12

2)1(

5

2arcsin2arctan

12

0

k

k

k

k

.10730091830127584519... 1

2 8

1

4 1

1

2 21

( )

( )

k

k k GR 0.237.2

= 1

16 1

2

1621 1k

k

kk

k

( ) GR 0.232.1

= ( )

1

4 60

k

k k

= 1

7 31 x x

1 .10736702429642214912... 3

28

.10742579474316097693... 1 44

5

1

4 1

1

1

log( )

( )

k

kk k

... one-ninth constant

.1076723757992311189... 10 62 21

ek

k kk ( )!

... e Ik

kk

0

12

20

1

2

( )

( !)

!

...

4 2

1

21 1

21

2coth

i i

= k

k kk k

k

k

kk

1

4

1

42 23

1

1

1

( )( ) ( ) 1

...

0

2

)!3(

2)1(

4

1sinh

8

1

k

kk

k

ee

.10819766216224657297... 3

2

3

2

1

2

1

2 22

log( )

( )

k

kk k k

.1082531754730548308... 3

16

1

12

2

0

( )k

kk

k k

k

... 432

9

1

3 2

1

21

Gk

k

k

( )

( / )

... 1

4 11k

k k ( )

1

1 .10854136010382960099... Hkk

k 2 11

...

161

14

2 2

0

2

ee x xx sin cos dx

... ( ( ) ) k

kk

1 2

22

... 27 4 3

481

13

15

logx

dx

x

.109096051791331928583... 71 6

108

1

31

2

21

2

0

1

k kx x

k ( )log( ) log x dx

...

12 )22(4

)2(

4

)3(7

4

1

2

log

kk kk

k

... x dx

x

6

120

1

1

=7

641

71

1

( ) kk

k

k

... 1

2

1

21

1 2

k

ke

k !sin sin sincos( / )

... 1

12

2

42

1

2 3

1

21

csch( )k

k k k

... 4 2 5

6

1

4 2

21

1

( )

( )

k

kk k

k

k

... ( ( ) ) k

kk

1

2

2

2

... 1

212

0

12

Ei e Ei e dxe x

( ) ( )

2 .1097428012368919745... 1

22 k kk log

=9

82

1

2 91 2 2 1 3

0

cosh log( ) ( ) logk k

k

e J943

.10981384722661197608... log( )

27

12

1

506 5

1

k

k k

dx

x x

...

4

2

2 1

2

20

4

arctan sin

sin

/ x

xdx

... 1

132 k

k

kk

log

... dxx

xx

1

032 )1(

arcsin

64223

... 3 3

4 122 2 3

12

4 32

( )log

( )

k

k k k

9 .1104335791442988819... 83

...

( )

( )

( )3

4 41

k

kk

Titchmarsh 1.2.13

...

2 21

2 1

2 2 2 1

2 1

8 2 11 1

( )!!

( )!! ( ) ( )

k

k k

k

k kkk

kk

=( ) ( )

1

4 3

1

4 1

1 1

1

k k

k k k J76, K ex. 108c

=

0

2/

12

)1(

k

k

k Prud. 5.1.4.3

= 11

2 30

( )k

k k

= dx

x40 1

Seaborn p. 136

=dx

x

dx

x

x dx

x20

20

2

402 2 1

1

= dx

x x( )1 12 20

1

= d

1 20

2

sin

/

.11074505351367928181... log( )

log2

3 3 31

1

3

1

12

k

k kkkk k

... ( )

1

430

k

k k

... 1

10 11

3

1031

1

1k

k

k

kk

k

( )( )

... 9 4 6 3 265

16 3

2

41

( ) ( ) ( )( )

k

kk

... 4

... log3 2

3

... ( )

!

k

k kk3

2

... 2

3

... e Lie e kk

k2 2

0

1 1

1

( )

=1

9 41 0

2

2

log

log(sin ) sin cos/x

xx x x

dx

= ( ) ( ) ( )k k

kk

kk

kk3 1 6 1 91 1 1

k

1

.1114472084426055567...

1

21

1

21

1

21

=1

21

2 21

1

2

cot

=1

2

2 1

232 1k k

k 1

kk

k

( )

1 .1114472084426055567...

2

11

2

11

2

1

=1

2

2 1

231 1k k

k

kk

k

( )

.11145269365447544048...

2 3 2

36

2

3

2

32

1

2

3

23

log log( )Li

=log

( )( )

2

21 2 1

x

x x

... 1

2

5

4

1

9 2 12 10

log( )

arctanh1

9 kk k

= Im logii

12

... 1

541 kk

1 .1116439382240662949...

2

3

2

1

2

3

2

13333

333

1 iiii

i

= 1 1

21 3 12 1

1

1

1k kk

k

k

k

( ) ( ) ) 1

= 1 1 3 3 11

1

( ) ( ) ( ))k

k

k k

... ( )

1

5 607 2

1

k

k k

dx

x x

... Li k kk

41

1

9

1

9 4

... ( )kk

15

2

... 2

1

2 1

23 2 22 1k k

k

k

1

k k( )

( )

.11207559668468037943... ( ) ( )

( )

2

32

2

16

16

16 11

2

2 21

k k k

kkk k

.11210169134154575458...

1

21 1

1

1 2 21

i i

k kk ( )

... ( )kk

k 2 111

... e

F

k

kk

1

1

/

...

1

2

232 )(

)4(

)2(

12

5

k k

kd

Titchmarsh 1.2.9

=

p p

p22

2

)1(

1

... 8

2 2

1 1

2

12 9

15

1e

e

x

dx

x x

dx

x

sinh sinh

... 1

2421 3 2 22 csc

= 1

48 2 22 3 2 21 22csc sec sin sin

=( )

1

24 22

k

k k k

... 21

41

/k

k k

... 3 3

32

1

2 2 30

( ) ( )

( )

k

k k

=

1

2

1

23 31

Li i Li ik

k

k

( ) ( ) cos

=

1

0

1

0

1

02221

dzdydxzyx

xyz

... Likk

k3 3

1

1

9

1

9

...

( )

( )

k

kk

2 1

2

... 1

22 1 1

1 1

2

2

0

1

log( )ee

e

edx

x

x

... 1 1

11

, ,

kk

k

... 3

43

4

3

1

3 1

1

1

log( )

( )

k

kk

k

k

1 .11318390185276958041... 1

18 7 6 5 4 3 20 k k k k k k k kk

.11319164174034262221...

log

4

3

.11324478557577296923... ( )jkkj

4 111

4 .11325037878292751717... e 2

.11356645044528407969... 5

361

5

61

51

1

log ( )k kk

k

H k

.11357028943968722004... ( )

11

2

k

kk k

.11360253356370630128... log

!

k

k kk 22

.11370563888010938117... 3

22 2

1

4 12 21

log( )k kk

J104, K ex. 110e

= 1

2 1 2 2

1

2 11 1k k k k k 2kk

k( )( ) ( )(

)

= k

k k

k k

kk

k

1

4

2 2

431 1

1 ( ) ( )

= ( )

12

2

k

k k k

= dx

x x

dx

e ex x( ) (2 21

21 1

)

.1138888888888986028...

| ( )|( ) k k

kk

1

9 11

.11392894125692285447... 616 1

40

3

21

e k k

x

xdx

k

k

e( )

!( )

log

.11422322878784232903...

9

112

7 2

2242 2

3 43 4 3 4

// /csc

csch

= ( )

1

842

k

k k

.11427266790258497564... 4 8 2 8 2 3 2 3 3 3 12log log log log log

= H

kk

kk 4 21 ( )

... Likk

k2 2

1

1

9

1

9

= 3log33

16 2

2

2

Li

... Ei Eik kk

( ) ( )( )!( )

1 1 21

2 1 2 10

=

2 2

1

2

3

2

3

2

1

4i si i HypPFQ( ) , , ,

=11

96

1

2 41

k k kk ( )( )

.11466451051968119045... 4 2 4 22

32

2 22

2

21

log log( )

k

kkk

.11467552038971033281...

1

24 2 6 2 66 6

2 6csch sech

sinh

= ( )

1

6 1

1

21

k

k k

.11468222811783944732... 2 2 5 4 21 2 1

2 4

1

1

log log( ) ( )!!

( )!

k

kk

k

k k

... 1

16

1

2

1

8

1

2

1

2 4

1 2

1

cos sin( )

( )!

k

kk

k

k

... 2 1tan

... 1

2

11

1

Ei e Eie

k

k kk

( )sinh

!

... J2 1( )

...

23 1

)1(

k

k

k

... 1

10

3

5

4

54

6

5 5

cot cot cos logsin

=

1

104

8

5 54

12

5

2

5cos logsin cos logsin

=

1

104

16

5

2

5cos logsin

=1

5 1 5 21 ( )(k k )k

.11524253454462680448... dxex

x

e

e

1

02)(

log1)1log(

... 5

... log

log log (log )( )3

22 3

2

3

1

22 3 2

1

2

1

2

7 3

8

Li Li

=log ( )

( )

log2

0

1 2

21

21

2 1

x

x x

x

xdx

... log

( )( )( )

2

6 1 20

dx

x x x 4

... 3

256

2

...

log log log2 63

2

2 3

2 31

16 2

1 x

dx

x

=

126

1

0

6 11log)1log(

x

dx

xdxx

... 1

2

1 5

2

1

2

2

2 1

1

2 22

2 20

log( ) ( )!!

( )!!

k

kk

k

k k J143

.11591190225020152905... 1

4

1

2

1

2 2 12 11

arctan( )

( )

k

kk k

.11593151565841244881...

11 1

1)12(

2

)(2log1

kkk k

kk

=

1 )12(4

)12(

kk k

k

=

kk kk

log 11 1

22

= cos xx

dx

x

1

1 20

... j3 J311

... 2

312

9

4

27

12

3

sin k

kk

= iLi e Li ei

2 33

33( ) ( ) i GR 1.443.5

1 .1165422623587819774... 1

3

1

4 8

4

40

cossin x

xdx

... 7 3

4

2

2

1

4

2

4 2 221

( ) log ( )

( )

k

kkk

... arcsinh

2 2

2 1

2 31

2

2 2 11

1

( )( )

kk

k

k

k

k

k

... 7

8

8

7

1

7

1

1

log( )

k

kk

k

H

... 2

21

2 2116

1

2

1

2 1 2 1

1

4 1

(( )( )) ( )k k kk k

J247, J373

... ( ) log

( )3 24 2

5

332

1

2 1

2

3 21

k kk

=15

128

1

3

4

1

( )k

kk

k

...

12

13

90

1

2 1 2 70

( )

( )(

k

)k k k

... 2

3

1

2

1

4 2 10

arctanhk

k k( )

... 1

12

3

2

3

23 8 3csc sec sin

=( )

1

4 1

1

21

k

k k k

... 2 1

2

1

3

2 1

2

1

2 3

2

1

G k

k kk

( )!!

( )!! J385

...

4

2112

4 2 2

( ) ( )( )

Lr k

kk

2 .11744314664488689721... 2 24 1

1

4( ) /

k

k

.117571778435605270... ( ) ( )

( )

( )3 2

4

43

2 21

H

kk

k

=2

17

... log( )

, ,9

8

1

9

1

8

1

91 0

1

1

1

kk

k

kkk k

= 21

172

1

17 2 12 10

arctanh

kk k( )

K148

...

1

1)2()2)(1(

2

1

k

kk

kk

... dxx

xdxxx

4/

0

34/

0

2

tan

sincossin

26

1

1 .1180339887498948482... 5

2

1

2

2 1

200

k

k kk

.1182267093239637759... 3

21 1

1

2 2 21

cos sin( )

( )!( )

k

k k k

... 11

21 1 1 1 2 2 ( ) sin (cos ) log( cos ) 1

=cos

( )

k

k kk

11

... G k

k

k

kk

log ( ) ( )

( )

2

4

4 1 2

16 2 11

= 21

22

1

41

kk kk

arctanh arctanh

.118333333333333333333 = 71

600

1

1 61

k k kk ( )( )

... 1 2

1 k

k

kk

log

.118379103687155739697... 7

5760

4

4 4

Li i Li i( ) ( )

...

3

1

6

5

432

1

36

)3(13

3162

5 )2()2(3

=

03)23(

)1(

k

k

k

... 1 11

e x ee x, dx

.11862641298045697477... 1 1 2 1 11 2 1

2 2 1

1

1

log( ) ( )!!

( )! ( )arcsinh

k

k

k

k k J389

= ( )

( )

1

4 2 1

21

1

k

kk k

k

k

.11865060567398120259...

H kk

kkk k

1

2

2

22

2 11

1( ( ) )

log

... 2 3 11

3

30

e Eik

k kk

( )!( )

... 2

8

1

4 12

20

1

log

( )

x

x xdx GR 4.241.11

...

4

2

3

1

2 506 4

1

4

0

4

( )

tan/k

k k

dx

x xx dx

... 8

1

22

1

2 1

2

0

1

logarctanx x

xdx

... k

kk

2

0 1( ) !!

... ( )

!

k

k

!!

k

1

0

.1189394022668291491...

4

33

4 31

3

2015

51

83

1

( ) log x

x xdx

x

e edxx x

.11910825533882119347...

5

36 144

2

18

2

0

1loglogx arccot2 x x dx

... 2 1 1

2

2

21

sin log coslog

e

x x

xdx

e

... 1

1

12

1

21 0e e

ex

edx

k

kk

x

( ) cos( )

... ( ) ( )4 5 2

... 3

8

.1193689967602449311... ( )( )

12 1

1 1

1

kk

k

k

... 4 4

.1194258969062993801... 2

21

8

144

1

12 9

G

kk ( )

... cossin ( )

( )1

1

2

1

2 2

1

1

!

k

k

k

k

... ( ) ( )( )

3 41 4

1

k

kk

... 1

4

1

2

1

2 4

1

1

sin( )

( )!

k

kk

k

k

... Ei

e

k

k

k

k

( ) ( ) ( )

!

1 1 1

1

1

=3

25 5 11

( )kk

k

.12025202884329818022... 1

2

1

2

1

2

4

5

1

2 22 11

arctan log( )

( )

1

k

kk k k

... 7 3 3

15

2 1

2 31

1

k

k kk

k

k

( )

( )

... ( )

( ) (3

12

1

2 1

2

21

)2

H

k k kk

k

... ( )

( !){},{ , ,},

11 1 1

30

k

k kHypPFQ

... e e ee

e

k

e k

k

kk

log( )( )

( )

11

1

10

...

2)1(coth

2122

1

i

k

H

k

k

= i

i i i4

1 1 1 12 2 1 1 ( ) ( ) ( ) ( ) i

... ( log )

arcsin log1 2

8 0

1 x x x dx

... 2 2

5

47

300

1

1 5

1

50

1

21

log

( )( )

( )

k k k kk

k

k

= log log( )11

11

64

0

1

x

dx

xx x dx

... 3

21

230

log( )k

k k

...

4 2 2 2

4 1 2

81

tan( ) (

)

k

kk

k

=1

2 1

1

8 1

6

2 1 8 12 21

2

2 21k k

k

k kk k

( )( )

.12078223763524522235...

log log log

2

1

2

4 3

2

=

1

2

1

1 20 e

dx

e xx x GR 3.427.3

=0

2

2

sinh

cosh

x dx

xe xx GR 3.553.2

.1208515214587351411... 3 3

212 2 10

1

1

1

3 31

( )log

( )

( )

k

k k k

.12092539655805535367... 1

642 kk

.12111826828242117256... 3

30256

1

4 2

( )

( )

k

k k

...

( )

1

2

1 2

1 22

1

k

kk

k

kk

... 1

6

1

2 2

14

2

2 2 21

2

coth

( )k kk

... log5

6

.1213203435596425732... 3

22

1

4 1

21

0

( )

( )

k

kk

k

k

k

k

= sin tan/

x x d2

0

4

x

... ( )2 1

4 11

kk

k

...

3

1

9

1

29

)2(4 )1(2 g

=log logx

xdx

x x

xdx3

0

1

311 1

... ( )3

2

... 43

2

3

2

1

2

1

21 3

1

2 2 60

log , ,( )

( )

k

kk k

... 14

153

41

( )( )

a k

kk

Titchmarsh 1.2.13

.12202746647028717052... 3

66 2

9 3

21

1

2 30

loglog

log( )x

dx

... 1

8 42

1

4

1

21

csch( )k

k k

=sin cosx x

edxx

10

...

03

31

)2(3

)(

12

3

1

9

34)3(26

k k

.122239373141852827246... ( ) ( )

1 3

9

1

9 1

1

13

1

k

kk k

k

k

1 .12229100107160344571... ( ) ( )

( )4 5 1

51

k

kk

HW Thm. 290

=

14

1

( )k

kk

.12235085376504869019... 12

21

2 22

log

( )k

k k k

2 .12240935830266022225... ( )k

k kk

22

.12241743810962728388... 21

4

1

2 42

1

1

sin( )

( )!

k

kk k

.12244897959183673469... 6

49

1

6

1

1

( )k

kk

k

1 .122462048309372981433... 2 11

6 11 6

1

1

/ ( )

k

k k

.12248933156343207709... 1

2

1

4

1

8

1

16 2 103 1

2

arctan( )

( )

k

kk k

dx

x x

... 2

2

0

18 2 11

361

1

loglog logx

xx dx

=log( ) logx x

xdx

14

1

.1228573089942169826... 7

21

1 2

1 2

0e

k

k

k

kk

( )

( )!

.12287969597930143319... loglog log ( ) log ( )

( )2 1

2

2

2

3

3

4

1

1

2 3 2

20

1

x

x xdx

.1229028434255558374... sin ( )

!( )

2 2

2

1 21

20

k k

k k k

... 325

48

15 1

12

51

( )

( )

k

k k

=8

65

1

2 81

1

3 2 1 2

cosh log

( ) ( ) logk

k

ke J943

4 .12310562561766054982... 17

1 .12316448278630278663... 3

2

1

3

1

3 2 10

erfi

k kkk

! ( )

2 .12321607812022000506...

1

221 )12(

2

2

)12(

2

1

2

1

24

1

kkk k

kkk

... Champernowne number, integers written in sequence

.123456790123456790 = 10

81 10

1

21

4

1

k kk

k

k

kk

( )

... 37

226803

62 ( ) 18 MI 4, p. 15

= 2

36

1

32 4

1

32 2

13 2

2

41

( ) ( ) ( ) ( ) ( )( )

H

kk

k

.12360922914430632778... 1

3

2

9 121 1

k

kk

k

k

( )

... 21

22

1

2 4

1

1

cik k

k

kk

log( )

( )! ( )

... 355

1082 4 3

4 31

( ) ( )( )

k

kk

8 .1240384046359603605... 66

... 3

2

4

3

1

3 12

1

1

log( )

( )

k

kk

k

H

k

...

2

1

41

21

2

1

2 4 121

i i

k kk ( )

=( ) ( )

1 2

2

1

2 11

k

k

1

k

k

529 .124242424242424242424 = 174611

330 20 B

...

2 41

1

4 coth ( ) ( )i i

=1

1

1 1

13 22

3 12

42k k k k k kk k k

... 4

251

4

5

1

4

1

1

log( )k

kk

k

kH

... 5

2

5

4 52

1

log( )

1

H

kk

kk

.124511599540703343791... ( )( )

18 1

1

1

kk

k

k

... 1

9 32

1

6 4 6 11

log( )(k kk )

J263

... 11 215

2 2 10

log( )(

3)

k

k kkk

... 8 22

312

1

2 1

2

2 21

log( )

k kk

... 1

512

1

163

1

4 1

2

40

1

, ,log x

xdx

6

... ( )

1

2 1

1

1

k

kk

k

... sinh sin cos( )

cosh( )/ /2

4

2

4 2

1

2

1

2

1 4 1 4

i

=sin /

( )!

k

k kk

2

2 41

.12500000000000000000 = 1

8 4 21

1 2

k

kk ( ) J364

=

1 )3)(2)(1(k

k

kkk

H

=

1

5

)1(kkke

k

= ( )k

kk 4 11

= x x x3

0

1

arcsin arccos dx

= log 11

21

5

x

dx

x

1 .125000000000000000000 = 9

83

2 H ( )

4 .125000000000000000000 = 33

8 3

3

1

kk

k

... 1 2 12 4 6 3 8 52

3

51

( ) ( ) ( ) ( )( )

k

kk

.125116312213159864814...

2

2 21

26

152

2

4

1 1

( )( )

( )

( ( ) )k

k nk n not squarefree

.12519689192003630759... 1

6

4

9 3 2

1

2

2

20

2

( sin )

( sin )

/ x

xdx

...

1

8192

1

8

1

16 22

30

( )

( )kk

... ( )

( )! (

1

1 2 2

1

1

k

kk k k )

Titchmarsh 14.32.1

1 .12522166627417648991... dxx

x

0

4

442

sinh)4()2(2

453

1 .1253860830832697192... 12 1

10

1

eEi Ei

e dx

x

x

( ) ( )( )

2 .12538708076642786114...

1

22

2

22log

6 kkkH

...

1

3456

1

6

1

12 102

31

( )

( )kk

... csc ( )3

6

1

18

1

9

1

2 21

k

k k

.12549268916822826564... 3

6

5

3

1

12

7

3

3

18 3 32

arctan log

dx

x x

... ( )k

kLi

k kk k4

24

1

1 1

. 1255356315950551333...

2 2

int2

11

)(11)()(1

k npowereger

nontrivialak

k n

kkk

=

S )1(

1, where S is the set of all non-trivial integer powers

... 1

22 2

7

2

1

4

1

33 3

5

2

1

42 1 2 1F F, , , , , ,

=

1

21

1

21

2)1(

)!2(

!!)1(2arccsch

125

54

25

4

k

k

k

k

k

kk

k

kkk

... 7

2 15 9459 3 3 5 28

1

1

2 4 6

6 31

( ) ( )( )k kk

... ( )k

kLi

k kkk k2

1

2

1

222

21

... 2

92 1

2

2 21

sinh( )

kk

...

1

1458

2

9

1

9 72

31

( )

( )kk

... Likk

k4 4

1

1

8

1

8

.12613687638920210969... dx

x x3 22

... 215

2 305 3 5

1

1

2 4

5 31

( ) ( )( )k kk

...

3

23

1512

7 3

128

1

1024

1

4

1

8 6

( )

( )( )

kk

...

3

2

2

31

3

1

1

13

20

1

log loglog

x x x

dx

... 1

2 3 3

2

3

1

3 506 3

1

log ( )k

k k

dx

x xdx

... 1 3 2 4 3 1 12

2

( ) ( ) ( ) ( ) ( )k

k

k k k

=4 2

1

3 2

2 31

k k k

k k

1

k

( )

... 11 3 1

6 2 1 12 110

dx

x /

... 7

2

... 11 4 e

... log 1 2 e

.1269531324505805969... 1

2

3

8 131 1

k

kk

k

k

( )

1 .12700904300677231423...

4

34

3 3

4

3 3

11

3 1

2

21

i i kk ( )

... Likk

k3 3

1

1

8

1

8

.1271613037212348273... 3 2

4 8

1

16 4 421 2

log ( )

k k

k

kk

k

= dx

x x x x4 3 21

.12732395447351626862... 2

5

3 .12733564569914135311... 2 22 1

1

2( ) /

k

k

.127494719314267653131...

( )2 12

1

k

kkk

.12755009587421398544... ( ) ( ) ( )( )

4 10 2 2 3 151

14 31

k kk

.12759750555417038785... 13 3

216

4

1296 3

1

432

1

3

1

6 4

32

31

( )

( )( )

kk

1 .12762596520638078523... cosh( )!

/ /1

2 2

1

2 4

1 2 1 2

0

e e

k kk

AS 4.5.63

= 11

2 120

( )kk2 J1079

.1276340777968094125... ( )

1

2 111

k

kk

...

( )

( )

4 1

2 1

... 4 26

1

2

1

2

2

3 21 1

log( )

k k

k

kk

k

=2

1

0

22 )(

x

dxxLi

... 21

4

1 1

2 42

1

arcsin( )!(

( )!

k k

k kk

)! K ex. 123

... 1

4

5

3 142

log

x dx

x

... 120

1

( )kk

k

... 1

4

2

4

3

12 1 2 32 2 20

dx

x x x( )( )( )

... k kk

kk

k

3

1

3

12 1 2

( )

... 2

1 ... sin ( ) cos ( ) ( ) sin1 1 1 1 1ci si

= x x x dlog sin( )10

1

x

1 .12852792472431008541... k

kk

k

k

k

2

41

1

11

1

21 4 2

( ) ( ) 1

... 2 2

3

1

31

13

14

loglog

x

dx

x

... 5

6

... 1

2 21

k kk

( )

... Jk k

Fk

k3

00 12

1

34 1( )

( )

!( )!; ;

LY 6.117

.12895651449431342780... 1

250

2

5

1

5 22

30

( )

( )

kk

.12897955148524914959... 1

81 ( )k kk

... 888

5 51

2 1

161

1

( )kk

k

k

k

k

.12913986010995340567... 1

8 21

kk k

... 12 1

1 k kk

... 27

23 1 3

1

9 5 31

log ( )

k kk

1 .12928728700635855478... 1

17 6 5 4 3 20 k k k k k k kk

... 2 1

2112

21

1

log( )

( )

k

k

k

k J347

= k

k k

k k

kk

k

1

4 2

1

23 21 2

( ) ( )

=log

( )

log

( )

log

( )

2

31

2

30

1

20

1

1 1

x

xdx

x x

xdx

x x

xdx

1

.12950217839233689628...

ee

k

k

k

1

41

/

.12958986973548106716... ( )3

12

1

4

2 12

6 5 42

3

k

k k k kk

= k k kk

( ) ( )2 22

1

2 .12970254898330641813... 1

1 1 13

0 ( !){},{ , },

kHypPFQ

k

3 .12988103563175856528...

iii

2

1

2

1tanh

.12988213255715796275... 13 33 2

2 1k k kk k

k k

( ) ( )1

.12994946687227935132... 1

6 CFG A28

1 .13004260498008966987...

Gx

x xdx

2 8

7 3

42

1 1

2 2

20

1( )log

arccos

( )( )

.13010096960687982790... 18

7 3

4

2 1

2 3

2

30

( )

( )

k

kk

... Hk

kk

3

1 3

.13027382637343684041... 1

4

1

2 20

sinh( )!

4

k

k kk

... 1

22 1

12

2

log( )

k

k kk

= 11

21

1

2

k

kk

kk

( ) log

.13039559891064138117... 101

12

31

3

k kk ( )

K ex. 127

... ( )

( ) ( )( )

k

k k k kmk m

kj j

jk mk

1

1

11

2 22 21

1

... 02 12

1 1( ) ( ) ( )k k jkk kj

=1

1

1

1 122 2k

k

k k

k

kjjk k k

( )

( )

( )

2

... 02

1( ) ( )k kk

...

4 32

2

2

22

0

4

log

tan/

x x dx GR 3.839.1

... log

( )

( ( )) logk

k k

k k

kk k2 2 2

22

21

1

=j k

k jkj

log2

21

... 3

4

1

4

1

20

e

e

k

kk

( )!

... ( ) ( )

log

1

2

1

3

1

21

1

2

2

2

k

k k

k

k kk k

k

... 1

8 22

1

420

csch( )k

k k

... cos cosh

( )

/ / 1 4 1 4

02 4

!

k

k k

... 2

2

0

1

44 6 G xarcsin log x dx

... i i( ) cos ( ) ( ) cos ( )/ / /1 1 1 13 4 1 4 1 4 3 4 /

... H

kk

kk 8 2

1

... 1

1 k kkk !

... 1

22 22 2 ( ) ( )( ) ( )i i

=x x

e edxx x

2

0 1

cos

( )

... 7 3

64

1

4 3 31

( )

( )

kk

... ( )( )

11

2

3k

k kk

4 .1315925305995249344... 1

22

( )kk

... log

arctan7

6

3

6

3

3

5

3 132

dx

x

... k

k k k k k ks d

k k

1 1 1

142

3 42 3

4

log log log( ) ) s

...

2

3 31 2

414

41

384

3

4

1

4

1

1 2

Gk

k

k

k

( ) ( ) ( )

( / )

... 4

2

21

1

2 2 2 30

log ( )

( )(

k

)k k k

= 12

( )kk

=

134

2/1

0

2 11log)41log(

x

dx

xdxx

... 4

2

2

1

2 2 1

1

1

log ( )

( )

k

k k k J154

=

1

2/

1

)1(

k

k

k Prud. 5.1.2.5

= xx

dxlog 11

40

1

= log( )

11

2 10

x xdx

=

12

arctandx

x

x

.13197325257243247...

ieLi e e Li e

k

k

ii i i

k2 132

3 31

( ) ( )sin

( )

3 .13203378002080632299...

3 3

2 2 3

1

3

13

13

log ( )

( )

GR 8.336.3

.1321188648648523596...

1

1

2)1(

3

2log1

9

2

kk

kk kH

.1321205588285576784... 1

2

1 1

2

1

1

e k

k

k

( )

( )!

... 6

7

7

6

1

6

1

1

log( )

k

kk

k

H

... 1

742 kk

.13225425564190408112...

( )5

21

kk

k

11 .13235425497340275579... 5815

729 33

6

0

ek

k kk

!

.13242435197214638289... log( ) 2

.13250071473061817597... 3

21

2 2

7

1

14

1

3 3 20

log ( )

( )(

k

)k k k

25 .13274122871834590770... 8

1 .132843018043786287416... ( )( )

12 1

1

1

kk

k

k

.13290111441703984606... 1 1

2 21

2 22 2 1

0e eek k

k

cosh( ) ( ) J943

.13301594051492813654... 2 24

3 2

23

1

2 1

2 2

2 21

loglog

( )

k

k k k

.13302701266008896243... 116 8

5

8

2

4 8

3

8

cot cot logsin logsin

= ( )

1

4 506 2

1

k

k k

dx

x x

1 .13309003545679845241...

4 2 2 20log

dx

x x

.13313701469403142513... ( )1 8

1 .13314845306682631683... ek kk

kk

k

8

0 0

1

8

1

2

! !!

.13316890350812216272... 1

2 2 5 5

1

5 1

1

21

csch

( )k

k k

.1332526249619079565... 2 2 2 22 1

2

1

2

log log( )(

H

k kk

k )k

1 .1334789151328136608... 3 5 2 3 41

( ) ( ) ( )

H

kk

k

.13348855233523269955... 5

12

2

32 1 2

2 1

4 2

1

1

log log( )

( )

k

k k k

k

kk

1 .13350690717842253363... ( ) ( )2 2 11

k kk

1

.13353139262452262315... log log8 71

8

1

811

Likk

k

= 21

152

1

15 2 12 11

arctanh

kk k( )

K148

... 3

161

3

4

1

3

1

1

log( )k

kk

k

kH

.13356187812884380949... ( ) log( ) log3

9

1 3

0

1

x x

xdx

6 .13357217028888628968... 2 1

1

k

k k kk

!

...

1

21

1 1

21

1 22

1

( )kk

k

1

=1

12 21 k kk ( )

... 7

2

2

7

2 1

7 2 10

arcsin( )

k

k kkk

.1338487269990037604...

1

0

2sin

10

52sin22cos

5

2xe

dxx

e

.1339745962155613532... 13

2

1 2 1

2 3

1

1

( ) ( )!!

( )!

k

kk

k

k

... 1

1 122 ( )

logk

k

kk

... 1

162 3 33 2 2 2 coth cothcsch csch 8

=1

12 31 ( )kk

... coth

( )

2

4

1

8

3

8 1

1

4

4

4 21

2

e

e kk J951

.134355508461793914859...

03 27327

2

x

dx

... arctan12

2e

dx

e ex x

... ( )

1

3 2

1

1

k

kk

... 1

11

12

1arcsinh

arcsinh2

2

kk

... e e

xdx

x4 1 4

0

... 2

2 2112

11

16

1

1

( )kk

J400

= 1

2

1

12 21

4 22k k k k 2

k k( ) ( )

= ( ) ( )k kk

1 2 12

1 .13500636856055755855... H

k

k

k

k

2

1 2

... cos ( )

( )

1

4

1

2 1

2

1

!

k

k

k

k

...

11

1

3

1

3

1

2

)1(

2

3log

3

1

kk

kk

kEk

LiH

1 .1352918229484613014... dxx

xx

1

0

2

1

log

9

49

3

2

... Ai ( )1

... 2 3 3 3cosh e e

.1353352832366126919... 1 1 2 1 2 1 1 2

20

1

00e k

k

k

k

k

3k k

k

k k k k

kk

( )

!

( ) ( )

!

( ) (

!

)

=( )

( )!

1 2

2

1

1

k k

k

k

k

1 .1353359783187...

( )

( )

k

kk4

2

...

21 2 2 11

1

2 1

cosh( ) ( )

k

k

k k

=cos

cosh

2

0

x

xdx

GR 2.981.3

.135511632809070287634... 1

41

1

4

1

2

1

4 1 2

0

e dx

e

x

x

1 .13563527673789986838... 9

... I2 1( )

... 5

14

2 1sinh

2

1sinh

2

1

4

11cosh

x

dx

x

... k

kk

!

!

10

=163

1200

1

1 51

k k kk ( )( )

... k kk

3 2

1

1 ( )

... 11 182 2

2

1

ek

k kk

( )!

... ii 11sinh2

=

12

1

22 22

)1(

1

)1(

k

k

k

k

kkk

=cos(log )

( )

x

xdx

1 20

1

GR 3.883.1

=cos(log )

( )

x

xdx

1 21

= dxee

xxx

0 )1

sin

.13607105874307714553... 16 23

4

1

41e k

k

k

( )

!( )k

... ( )k

kLi

kk k2

22

2

1

... 2 4 21

2 24

3

0

1

( )log

( )( )

Lix

x xdx

1

.13608276348795433879... 1

3 6

1

8

2

0

( )k

kk

k k

k

... 1

8 11

3

831

1

1k

k

k

k

kk

( )( )

=

4

33

)33(6

33

4

33

)33(6

32

3

2log

3

1

6

i

i

ii

i

... 4 11

1 41 4

1

ek k

k

/

( )!

...

0 0 24

1

4)!2(

!!

4

1arcsin

1515

16

15

16

k k kk

k

kk

kk

... 2 25

4

1 1

1 232 0

log( ) ( )

( )( )(

3)

k

k

k

kk k k k k GR 1.513.7

=

2

1

2

1)1(

)1(2

1

1

1

2k

k

k

kk

Lik

...

2122

)()1(111

1

kk

k

k

k

kk

... 11

1 1k

k k

( ) ( )

... 2 1

2

2 1

2

1

2 2

2

1

( )!!

( )!!

k

k kk

J385

= ( )( )!!

( )!! ( )( )

12 1

2

4 1

2 1 2 21

1

3

k

k

k

k

k

k k J391

.13675623268328324608... 13 3

27

2

81 3

1

54

2

3

1

3 2

32

30

( )

( )( )

kk

... 5

4 2 6

5

2

1

4

2 2

4 21

coth

k

k kk

= ( )( ) ( )

1

42 2

1

k

kk

k k 2

... k

kk

k

k

k

3

51

1

11

1

21 5 3

( ) ( ) 1

... ( )

( )3

26

1

1

2

2 31

k kk

... 1 33

4

1

4 11

log( )k

k k k J149

...

22 1

21

3

3sin

k k

.13707783890401886971... 2

2172

1

6 3

( )kk

=1

12 3

1

12 92 21 ( ) ( )k kk

= log 11

61

x

dx

x

...

( )

( )

( ) log3

32 31

k k

kk

... 9

2

1 .137185713031541409889... 2 2 2

212

2

2

2

28 2

7 3

2 2

log loglog

( )

( )

H

k kk

k 1

...

( ) ( )

( )

3 5

3

... ( )

1

232

k

k k

... 1

2 22 2 1

0sinh( )

e k

k

AS 4.5.62, J942

... 4

5

2

5 5

2

5

4

5 1 5 20

csc /

x dx

x

... ( )

1

4

12

1

kk

kk

F

... e

e xx1 4

02

2/

cosh

dx

12 .13818191912955082028... 2 1 3 1 12

0

log x dx

... log log 11

111

1

ee ek ek

k

= ( )k

e kkk

1

1 .13830174381437745969... I e

eF e

e

k k

k

k

2

0 10

2 1

23

2

; ,!( )!

1 .13838999497166186097... 1

11 k k

k

1 .13842023421981077330... 5

81 3

3

361

1

3 1

1

3 2

1

3 3

3

3 30

( )( )

( ) ( ) ( )k

k k k k 3

.1384389944122896311... 2 212

422

0

1

log ( ) log

K k

k

dk

k GR 8.145

... 9 3 9 3

4

13

231

log ( )!

( )!

k

k kk

... log 2

5

.13867292839014782042... 4 2 2 2 31

16

2

1

log log( )

kk k

k

k

=( )

( )!

2 1

2 41

k

k kkk

!!

... 2 21 12 1

4 1

2

2 21

Gk

k k

k

k

log( ) ( )

( )

.1388400918174489452...

16 2 1640

dx

x

.138888888888888888888 = 5

36

1

2 3

1

51

1

1

k k k

k

k

k

kk( )( )

( )

= k

k k k kk

2

1 1 2 3( )( )( )( 4)

2 .13912111551783417702... e pf k

kk

( )

20

... 14

5

1

2

1

2 1

1

1

arcsinh( ) !

( )

k !

!k

k k

k

... 2 1 2 1 2 2 2arcsinh log log 2

=( ) ( )!!

( )! ( )

1 2 1

2 2 1

1

1

k

k

k

k k k

.13920767329150225896... 1

31

2

4 2 16

2 2

20

4

log tan

cos

/

Gx x

xdx

2

GR 3.839.4

... 1

6 2 6

3

2

1

2 3

1

21

csch( )k

k k

.13940279264033098825... 2

1 12

1

erf e dxx( )

... e

e x

dx

x x

dx

x4

5

41

1 1

2

12

17

14

sinh sinh

... )3(2

5

2 .13946638410409682249...

4

1,

2

3,2,2

3

4

39

412 F

=

11

2

12)!12(

)!(

kk

k

kk

k

k

... sec ( )( )!

1

21

2 40

2

k

k

kk

E

k

... log log2

3

3

12 42 2

2

H kk

k

3 .13972346501305816133... log e e

.1398233212546408378... 1

5

1 2 1

5 2 20

1

cos sin sin

e

x dx

e x

=7

50

=1

2 71

0

2 1 7

cosh log( ) ( ) log

k

k

ke J943

... 1

4

... 1

121 e k

k

Berndt 6.14.1

... 5

62

1

2 1 2

1

41 0

log( )

( )

k k kk

k

k

= ( )( )

11

22

kk

k

k

=dx

x x5 41

... 31

3

1

3 2 10

arctanh

kk k( )

=9

64 91

kk

k

... ( )9 ... e i i 2 Shown transcendental by Gelfond in 1934

...

2 2

2

1

sinh( )

k

k

k

k !

... e

.14111423493159236387... 1

2 221

kk

k

.14114512672306050551...

1

8

1

1 1 1 13 3 3 3

3

1

logcos

e e e e

k

ki i i ik

...

Lik k

k

kk

k

k2

22

2

2 2 2 1

( )

7 .1414284285428499980... 51

... 2

32 2 4 3 4

1

2

22

2

2

12

30

1

log loglog

( )Li

x

xdx

... 3 2 32 3 1

1

1log log ( ) /

k k

k

... 323

2

77

6

1

2 50

log( )

( )

k

kk k

.1415926535798932384...

3 16

1

( )sink

k

k

k

1 .1415926535798932384...

2 14

1

( )sink

k

k

k

= arccos2

0

1

x dx

= x x d2

0

2

sin/

x

3 .1415926535798932384...

= 1

2

1

2,

=3 1

41

8

4 1 4 31 1

k

kk k

kk k

( )( )( )

Vardi, p. 158

=dx

x

e dx

e

x

x1 12

2

/

=x

xdx

10 sin

=dx

x2 20 cos

= log log1 13

12

02

0

x dxx

dx

= log( )1 22

0

xdx

x GR 4.295.3

=

02

22

1

)1(logdx

x

ex

= log22

0

121

1

x

xx dx GR 4.298.20

=

02

11loglog dx

xx

=

02

22

1

))1(log(dx

x

ex

= 1

0

2logarccsc dxxx

.1415926535798932384...

3 2 23

2

1

2

222 1

1

Fkk

k

k

k

, , ,

= 122214

15625 6

6

1

kk

k

.141749006226296033507... log ( ) log2 2

0

1

1 x x dx

... log log2

2 32 4

2

8

2

=

40

4

x x x dtan tan

/

x GR 3.797.1

...

e

B

kk

k

k

1 0 ! J152

.14189705460416392281... arctan( )

( )

1

7

1

7 2 12 10

k

kk k

=

1

21

)2(

2arctan)1(

k

k

k

[Ramanujan] Berndt Ch. 2, Eq. 7.5

= arctan( )

1

2 3 21 kk


... 7 3

2 28

1 1

2

2

12

( ) ( )( ( ) )

k k kk

k

.142023809668304780503... ( )( )

17 1

1

1

kk

k

k

... 4

1

43

4

2 22 1

2 41

log

( )!!

( )! ( )

k

k k kk 1

... 4

11

.142514197935710915709... 6 42

4

2

46

1

2

21 3 2

4

2 3 4

4

( )log log ( ) log

Li

= log ( ) log3

0

1 3

1

21

1

x

xdx

x

xdx

.14260686462899218614...

1 2

)()(

kk

kk

1 .14267405371701917447... 2

22 2

125 5

1

5 1

1

5 4csc

( ) ( )

k kk

... 8

1

4 12 21

dx

x( )

.1428050537203828853... 14

81

2

27

2

31

3

1

log ( )k kk

k

H k

2

... ( ) ( )

1 1 1 12

22

2

k

k k

k

k kLi

k

=1

7

...

3G

... 3

216

... ( )

logk

k k kk k

1

1

1 11

1

32

2 k

... ( )k

Fkk

2

.14384103622589046372... 1

2

4

3

1

7

1

7 2 12 10

log( )

arctanhk

k k K146

= dx

x x

dx

x x x32 0 1 2

( )( )( 3)

... 5

2

4

14 2

e

x

xdx

cos

( )2

... e e ek

ke e

k

i icos cos(sin )cos

!1

0

11

2

GR 1.449.1

= )1sinh(cos)1cosh(cos)1cos(sin

.14430391622538321515... 4

2

0

x e x dxx log

3 .1434583548180913141... log e e

1 .14407812827660760234... Li2

5

6

1 .14411512680284093362...

1

2

2

)14(

18

4222log3

k kk

kG

= ( ) ( )k k

kk

1 1

41

.14417037552999332259... Likk

k4 4

1

1

7

1

7

... 23

2

2

3

1

2 1

1

1

log( )

( )

k

kk

k

k

.14430391622538321515... 4

4 2

0

e edx

xx x GR 3.469.3

=

e x x dxx 2

0

log

3 .14439095907643773669... 2 2

2

1( )kk

.14448481337205342354... e e ek

k ee e

k

kk

1 1

1

1

11

1/ / ( )

( )!

.14449574963534399955... H

kk

kk

( )

( )

3

1 3 2 1

2 .14457270072683366581... 2

41

k

k k k!

.144729885849400174143...

12 )2(4

)2(loglog1

kk kk

k

e

...

0

2/1log

2

1)1(log

kk

k

k

k

k Prud. 5.5.1.17

... 31 5 32 53

2

11

25

1

( ) ,( )

kk

... 1 1 1 2

2 221

1

1k k

k

kk

k

k

cos

( ) ( )

( )!

...

2

3 22

1

16

3

2

11 2

k k

kk

k

k

( ) ( ) 1

=H

k k

k kk

kk

k2

2 1

2 2

4

2

( ) ( )

=4

1 12 2 20

x

x edxx( ) ( )

= log( )1 2

0

1

xdx

x GR 4.29.1

...

2

26

1

21 1

( ) ( ) ( )k

k

k k 2

... 2 3/

... 381

2187

1

2

1 7

1

( )k

kk

k

... 2 112

21

log( )eke k

k

GR 1.513.3

.14544838683609430704... 1

25

3

58 25

8 5

5

1

25

7

51 2

21

( ) ( )

=1

5 2 21 ( )kk

... Likk

k3 3

1

1

7

1

73 0

1

7

, ,

... 2 3 2

12 32 18

3

2

1

3

3

162

2

3 4

7 3

4

GGlog

( )

= H

kk

k ( )4 3 21

...

0

2coslog442arctan85log3100

1

25

11dxxxex x

... ( )

!( )!

k

k kk I

k kk k

1

21

1

221

1

=7

48

1

4

1

1 5

1

421 0

1

23

k k k k kk k

k

k( )( )

( )

= x x d3

0

1

1log( ) x

= log 11

15

x

dx

x

... )2(56)4(140)6(56log24

53

=

1 4

1)2(

k k

k

... log ( )!!

( )!!

2

2

1

4

2 1

2

2 2

2

1

2 22

2

G k

k kk J385

... 4

... 2 22

21

1

2

2 1

1

loglog ( )

k

k

k

H

k

.1463045386077697265... 44

2 21

2 1

2

21

log

( )k kk

GR 0.236.7

= log( ) log1 2

0

1

x x dx

... 2 21

2

3

22

1

2 1 2

1

1

arctan log( )

( )

k

kk k k

... 1

842 kk

... cos

/

x

edxx 2

0 1

... 4

22

22

1

2

1

... 1

2

2 1 1

2 21

cos sin log sin log

e

x x

xdx

e

= xe x dxx sin0

1

1 .14649907252864280790... 1

21 k kk !

2 .146592370069285194862... 6

52

3 20

2

cos

/ cos

x

xdx

... 24 13 1 20 11

2 2

1

1

cos sin( )

( )!( )

k

k

k

k k

... 1

121 ek

k/

Berndt 6.14.1

.14711677137965943279... )2()2(4

)1()1( iii

=

222

1

1

11)12()1(

kk

k

k

kkk

=

0 )1(

sin

2

1dx

ee

xxxx

.1471503722317380396... ( )

1

2 2

1

1

k

kk

...

0

6

3)cos(sin

10

1

40

3log9dx

x

xxx Prud. 5.2.29.24

.14722067695924125830...

loglog log

2 2

2

2

2 122 3

1

2

Li

= log log log2 3 21

2

1

42

2

Li

= log( )1

20

1

x

xdx

... 90

90

1

4 14

( )

.14726215563702155805... 3

643

0

1 x xarccos dx

... 11 1

111

Ei

e

H

kk k

k

( )( )

( )!

= dxxxex 1

0

1 log

.14747016658015036982... ( ) ( )( )

( )

1 3 11

11 2

1

3 3

3 32

k

k k

k kk k

k

... 1

9 1

3

931 1k

k

kk

k

( )

... 58 1

2 30

e k

k

kk

( )

( )!

.14779357469631903702... 2

2

1

21

arcsinh

=( ) ( )!!

( )! ( )

1 2 1

2 4 1

1

21

k

k

k

k k

1 .14779357469631903702... 2

2

1

21 2

0

1

arcsinh

x dx

.147822241805417... ( ) log

1

122

k

k

k

k

.14791843300216453709... 3

23 1

1

9 31

log

k kk

J375

= ( )2 1

91

kk

k

... 4 31 4

1 4

0

ek

k kk

/

( )!

1 .14795487733873058... H5 23

/( )

.14797291388012184113... 2

42

1

4

1

22 21

coth

kk

1 .14812009937000665077... 1

7

2

8 2

1

4 421

tan

k kk 7

... ( )k

kkk 21

=4

27

1 1

20

( ) ( )k

kk

k

... ( )

( !)

k

k kI

kkk

2 12

12 2 0

10

... Likk

k2 2

1

1

7

1

72 0

1

7

, ,

... 4 21

2

10

3

1

2 2 50

arctan( )

kk k

... 3

27

... 13

18

3

2

1

3

1

3

1

3

1

3 21

loghg

k kk

=( ) ( )

1

32

k

kk

k

=dx

x x x4 31

2

... 1

4

3

4

1 2

220

e k k

k k

k

( )

!( )

... 21 5/

... 16 7 4 31

16 1

2 2

0

( )( )

kk k

k

k

... 41

22

23

2

3 21

Lix

x xdx

log

.14900914406163310702... 21

2

1

3

1 2 1

2 3

1

1

log( ) ( )!!

( )!

k

kk

k

k k

... logk

k kk3 2

2

... 4 7

9

3

2

3 2

1

e k

k

k

k

( )!

... 93

2

7

2

1

2 1 20

log( )

( )( )(

k

kk k k k 3)

=

0 )2(3

1

2

1

kk k

... 8

19 42 2

0

2

ee e xx

/ sin cos

x dx

...

1

13 !

3)1(

3

k

kk

k

k

e

6 .14968813538870263774... 2 2 33

2

1

4

1

2 12 11

Fk k

kk

, , ,!( )!

( )

!

... 19

5 23 2

1

4 1

1 1

421

2

log

( )

( ) ( ( ) )

k k

k

k

k

kk

1 .14997961547450537807... 2 13

2

k

k

k ( )

.15000000000000000000 = 20

3

.150076728375356585727...

22 2

11log

2

1

)1(2

1)()1(

kkk

k

kkk

k

.15017325550213874759... 2

3

3

21

3

4 21

sin

kk

...

1

2

1

13 )1(

)1(

)2(

1

8

)3(

k

kk

k k

H

k

= dxx

xxdx

x

x

1

0

1

0

22

1

log)1log()1(log

= 1

0

1

0 1

)1log(dydx

xy

xy

... 24

)21log(228

5

8

1

16

1 2)1()1(

Berndt 3.3.2

= 12(21(2 22 iLiiLii

= 2

2 2 1

2

20

k

k

k

k k

( !)

( )!( )

= 22 2

0

2 x x

xdx

sin

cos

/

... 1

21 1

1

2 11

sin cos( )

( )

k

k

k

k !

= sin1

2 51 x

dx

x

... 3 3

4 15

2

42 2

2

4

4 2 23

4 ( ) loglog

log

6

1

2

21

43 24Li ( ) log

=log ( )

( )

3

20

1 1

1

x

x xdx

... 8 2 4 3 11

3 11

log log( )

( )

k

kk k k

... 1

4 2 6

2

3

1

3 2

1

21

csch

( )k

k k

... 1)1(1)(2

kkk

...

03 3

)1(

k

k

k

...

1

2

2 1010

10,2,

10

1

729

110

kk

kLi

...

12

2

)4)(2(36

113

3 k kk

k

1 .15098236809467638636... 3

10

3 3

10

2 2

5

1

6 5

1

1 6 121 0

log log

( )(k k k k )k k

= dxx

x

1

06

6 )1log(

.15114994701951815422...

12 3 32 20

dx

x( )

...

127k

kk

k

H

.1511911868771830003...

( )4 1

2 14 11

kk

k

81 .15123429216781268058... 8

.15128172040710543908...

1

01

3

)()1(

kk

k k

1 .15129254649702284201... 10log2

1

.15129773663885396750...

13

1

k kkF

.151632664928158355901...

1

21

2!)1(

4

1

kk

k

k

k

e

1 .15163948007840140449... ( ) ( )

( )

3 5

4

2 .15168401595131957998...

1 22/52/5 )1(

1

k k k

k

k

k

.151696880108669610301... 25

4

2csch

162coth

82

2

=

222

2

1

1

)14(

4

4

1)2()1(

kkk

k

k

kk

.15169744087717637782...

82 2 1 2

0

2

( log ) log(sin ) sin/

x x dx GR 4.384.9

.151822325947027200439... 1

2log

e

.15189472903279400784...

1

2

1

32

)1(

3

2log

69

4

k

k

kk

=4

12

11log

x

dx

x

1 .151912873455946838843... 1

16 5 4 3 20 k k k k k kk

.1519346306616288552... 5

6

6

51

51

1

log ( )

k kk

k

H

... log ( ) k

kk2

2

... 2 5 21

22 1

1

1

log1+ 5

2

( )

( )

k

k k

kk k

...

1

2)12(

61

2

3cos

5

1

k k

...

1

22

)12)(1(3

2log2

3

2log4

18 k

k

kk

H

.1523809523809523809 =)4)(3(4

12

105

16

0

kkk

kk

k

...

1 87

8log

7

8

kkkH

...

0

2

)7)(4(!

2

42

1

k

k

kkk

e

... 2)1(

1 .15281481352532739811...

1 3!)!12(

!)!2(

3

1arcsin

4

23

2

1

kkk

k

...

21

1

kkk

k

.15289756126777578495...

1

2

1

2

3

5

)1(

)1(dx

e

x

e

k

ee

eex

kk

...

13

234 )1(

11)3()2(3

kk kkkk

=

1

1)3(k

k

...

2

2)2()()3()2(4k

kk

...

3

1

3

1

2

1

3cot

322

1

=

113 3

)12()2(

3

1

kk

k

kk

kk

k

... )3(

)3(log

... 2 2

10

1

six

xdx

arcsin

arccos

... )1(Im2)1()1(1coth iiii

=

1

11

k ikik

...

12 1

21coth

k k J947

... coth ( ) ( ) Im ( ) 1 2i i i i

... 1 2

2

1

2 1 2 2 1

1

1

log ( )

( ) (

k

)k k k k GR 0.238.2, J237

= kk

kk

log2 1

2 11

1

J128

=

1 )48(

1

k kk

= 1

2 22

kk k k( )

= ( )

( )!

2 3

22

k k

k

!

k

= ( )

1

2 40

k

k k K Ex. 109b

= ( )

1

8 2

1

31

k

k k k

=

12

2

1 4

11log41

)12(4

)2(

kkk k

kk

k

= dx

x xx dx5 3

1

3

0

4

tan/

=

02

3

sinlogsin1

sindxx

x

x Prud. 2.6.34.44

= log

logx

xdx

x

x

dx

x21

2

2 1

=

1

02

12 )1(

)1log(

)1(

11log dx

x

x

x

dx

x GR 4.291.14

= ( ) arctan10

1

x x dx

= x

dx

xx log2

1

log2

1

1

11

02

GR 4.283.5

=

1

2

1 2

0 xe

e

x

dx

xx

x /

GR 3.438.1

= dx

e ex x2 20 1( )

1 .15344961551374677440...

0

2

4

0211 )!()1()2()2()2(3)2(2

k

k

k

kJJJJ

=

1

1 2

)!12(

)1(

k

k

k

k

k

= 13

2

... k

k kk

2

2 11 1

.15354517795933754758... )7()1(

.1535170865068480981...

1

1

13 7

)3()1(

17

1

kk

k

k

k

k

... 7/1e ... 8/1

... )1(360)2(6)3(42

2log

590

181

=

2 4

1)(

k k

k

.15409235403694969975...

1

22

2

1

2

)112(

12

12

)2(

32csc

3sin3

48 kkk k

kkk

.1541138063191885708...

23

3

1

)1(

2)1)()(2)(1()1(

4

9)3(2

kk

k

kkkk

= 3)2(

=

134

2log

xx

x

= x x

xdx

2

0

1

1

log

= x

e edxx x

2

20 1

.154142969550249868...

1

)3(

)1(4kk

k

k

H

.15415067982725830429...

11 7

1

7

16log7log

kk k

Li

=

0

12 )12(13

12

13

1arctanh2

kk k

K148

.15421256876702122842... 2

2 2164

1

2 1 4

(( ) )kk

J383

15 .15426224147926418976...

0 !k

ke

k

ee Not known to be transcendental

=

0

!/1

k

ke

... 2 12

1 2 11 2 1

12 2

k

kk

k

k

k kk k

( ) log( )

... 2

... 3 10

... ( ) log ( )

1 21

1

k

k

k 1

... 2/3)2(

2

39

40H

... ( )

log

13

2

k

k k k

...

1 2 ),2(

1

k kkS

... 7

32 2

1

4 2 1

23

1

( )

( )

k

kk

k

k

k

k

.15468248282360638137...

13 1)13(

1

k k

=

6

35332

6

3534

3318

1

6

3log

3189

ii

i

i

.1547005383792515290... 2

31

1

2 1

21

0

( )

( )

k

kk

k

k

k

k

=( )

( )!

2 1

2 41

k

k kk

!!

...

0

2

16

1

3csc

3

2

kk k

k AS 4.3.46, CFG B4

=

122!)!2(

!)!12(1

kkk

k J166

.15476190476190476190 =

0 )103)(13(

1

84

13

k kk K134

... 3 61 3

0

1

e ex /

dx

...

1

4

)!1(13

k k

ke

...

1 !

)1(3

k k

kke

.15491933384829667541... 1

5

3

51

2

61

1

( )k

kk

k

k

k

... )(Re i

...

3

2

...

122

2 11 2

k kkH

... )3()2(

)3()2(

... )3(2log723log54318216 2

=

1

1

134 6

)3()1(

6

1

kk

k

k

k

kk

... dtt

tdx

x

x

e

0222 1

cos2

)1(

cos

...

2 202

121

1

)!(

1)()1(

k k

k

kJ

kk

k

.15580498523890464859... 4 4 2 22

3

1

22

arctan

1

2log Li

=( )

( )

1

2 2 1

1

21

k

kk k k

.1559436947653744735... cos ( )( )

2 12

20

!

kk

k k GR 1.411.3

.15609765541856722578... dx

x x4 31

x

...

kk

kk

kk

k

kk

11log2

11log221

4

1

1

1)( 2

122

.15622977483540679645...

1 2

)(

kk

k kH

.156344287479823877804...

7

10 4 21

2 1

41

1

coth ( )

( )kk

k

k

=

222 2

1)(Re14

1

k

k

k

ik

k

.15641068822825414085...

dx

x

xdx

x

x

e 1

3cos

)1(

3cos2

0223

.15651764274966565182...

1 0

1

221

22 !

2

1

11

2

11coth

1

1

k k

kk

kk k

B

kee

1 .15651764274966565182...

022

022

2

1

11

2

1coth1

1 kkk kee

e

J951

...

1

1

)2()!2(

)1(

2

231sin101cos6

k

k

kk

1 .15654977606425149905...

1

22cos

!

sin

42)2cos(sin

2

122

k

ee

k

keeeee

ii

...

3 3

0

6 x x dsin x

... 2

11

2 2 20

log

log log

ex

e xdx

... 3

6

3

41

13

13

logx

dx

x

...

05520 xx ee

dx

... 1

33 e

3 .157465672184835229312...

0

2

2

1

)3log()31log( dx

x

x

1 .15753627711664634890...

13

133 )13(

1

)13(

11

27

)3(26

kk kkG

1 .15757868669705850021...

4

3

2 1

2

0

2

x dx

ex

... 1

9

2

6

1

3 1

21

21

( ) logx x

x xdx GR 4.233.3

... cos ( )( )!

3

0

2

11

41

3 3

2

k

k

k

k GR 1.412.4

.15779670004249836201...

2 21

kk

k

...

1

21

/12 !)1(

1

kk

k

k

k

e

... log8

.157894736842105263 = 19

3

.15803013970713941960... 1

4

1

4

14 9

1

e x

dx

xcosh

.1580762... 1

1 2

1( )

int

a non trivialegerpower

...

12

21

02

2

)1(

log

)1(

log)2(

2

)3(3

xx

xdx

x

xx

...

1 )12(3kk

k

k

H

...

0 1 ),2(

!

k kkS

k

...

1

2

1

14

)1(

2csch

42

1

k

k

k

.158747447059348972898...

log ( )( )

22

22

12 1

41

1

i i k

kk

kk

...

1

2

1

3

)1(3csch

6

3

6

1

k

k

k

J125

...

14

12

2

1

log

)14(

1

216 x

dxx

k

G

k

...

1 )2)(1(242log6

kk kk

k

... 6 21

13

230

log( )

( )( )

k

k k k

...

1

2

22log68

kk

kHk

40 .15890107530112852685...

1

2

k kF

k

1 .1589416532550922853... B

k

k

kk

k

2

0 2

2

( )!

3 .15898367510013644053...

1 1 1 111 1 11 1

0

1

4

2

1

12

1

12

)(

2

)(

i j k liijkl

j kijk

i jkj

kk

kkt

... 3;1;)!(

3)32( 1

020 F

kI o

k

k

.15909335280743421091... 1

0

6 arcsin245

16

14dxxx

GR 4.523.1

.15911902250201529054... 5

2

1

21

1

2 2 1 2 1

1

2 11

arctan( )

( )(

)

k

kk k k

...

1 )2)(12(

1

12

2log1613

k kkk

.1591484528128319195... ( ) log arctan

18

1

94

9

82 2

1

2 21 2

1

k kk

k

H

... 21

... )5(

)3(

...

1

2

2

)22()2(

kk

kk

.159436126875267470801... 2 2

132

5

12

2 2

3

2

3

1

3

log log ( )

( )

kk

k

H

k k

1

.15956281008284... 7

96

1

2

1

161

2

24

24 1

4

21

Li e Li ek

ki i k

k

( ) ( ) ( )cos

=

1 )4)(1(

1

144

23

k kkk

...

0 )12(

1

4

1cothlog

2arccoth

kk ke

eee

.159868903742430971757... log log ( )

log ( )2

1

22 1

22

2 1

k

kk

k

k

k

k

k

... ( )1 5

6

=

1

1

4

)1(

25

4

kk

k k

.1600718432431485233... 1

54 5 1

41

( log log ) ( )( )

kk

k

k

.16019301354439554287...

112

1

2 76

)1(

6

1

kk

k

kk

k

k

H

kLi

.16043435647123872247...

1 )3(2

1

3

2log7

9

16

kk kk

... dxx

x

e

023 3

cos

32

... 24 2 7

601

14

0

1loglog

x

xdx

.1606027941427883920... 25 1

30

2

0

1 2

21

e k kx e dx

x

xdx

k

k

x( )

!( )

log

=( )

( )! !

1

1 20

k

k k k

...

1 )12)(12(

2cos

2

1

4

1sin

k kk

k GR 1.444.7

... 22

22log

2

242log

2

2

=

12

1

4

)1(

k

k

kk

=2

14

1

0

4 11log)1log(

x

dx

xdxx

... 2

22

2

2

2 2

2 21

14

0

1

log log log

xdx

...

11

1

2

log

2

2log

2

1

2

1)1(

kk

kk

k kLi

k

.1613630697302135426... 1

2

1

21

1 2

22 2 121

2

1

coth( )!

k

B

kk

kk

k

.16137647245029392651...

222

2

1

)!2()!2(

1)2(00

1 kJ

kI

kk

k

k

.161439361571195633610...

1

2122

)!2(

2

2

1log1sinhlog

k

kk

kk

B

e

e

= 2

2

2 12

1

kk

k

B

k k

( )!

=

12

1 )2()1(11log

kk

k

k

kii

1 .161439361571195633610...

1

2

!

2)1(

2

1log1sinhlog1

k

kkk

kk

Be Berndt 5.8.5

... 3 2

20

1

192 1

arctan x dx

x

2 .16196647795827451054...

1 2k

k

k

kk

1 .162037037037037037037 = 3)3(

216

251H

.16208817230500686559...

1

1

3)!2(

)1(1

3

1cos

kk

k

k

... 3 3

2

11

65

60

( )( )k kk

.162162162162162162162 =

0

6log)12()1(6logcosh2

1

37

6

k

kk e J943

= dxe

xx

0

6sin

...

112

2

2

)2(

)12(

122log4

2 kk

k

kk

kk

3 .162277660168379331999... 10

1 .16231908520772565705...

1 21 )!(

11,0,

2

322)1(

k k

eerfe

.162336230032411940316...

1

3

)!3(2

7113

k k

ke

...

15

21

4

)(6

)5(93

k k

k

... 2/12

12

3

1 22212

1

!

)()1( kekk

kk

k k

kk

k

...

1

3 )12(2)12(27

)1(

812

k

k

kk

= dxx

xx

1

022 )1(

arcsin GR 4.512.7

.16268207245178092744... 2 2 31

8 4 31

2

log log dx

x x

.16277007036916213982... 2 31

22

4

5

1

2 2 1 2 1

1

2 11

arctan log( )

( ) ( )

k

kk k k k

... 3

1

.162865005917789330363...

1

0222

22

1

)1log(

2

1

2

1

8

2log

96

11dx

x

xxiLi

iLi

...

3

2 2 2 23 3

256

1

1024

7

8

5

8

3

8

1

8

( ) ( ) ( ) ( ) ( )

= ( )( ) ( ) ( ) ( )

11

4 1

1

4 2

1

4 3

1

4 43 3 30

k

k k k k k 3

...

4

2

0ex e x x dxx

sin cos

...

( )log ( )

k

kk

k

1

2 1

.1630566660772681888... 7 3 12 1

2

1

e Eik

k kk

( )( )!( )

... ( )

( )

k

kk

1 2

2

...

1 849

8

kk

k

... 2/

0

322

cos27

40

6

dxxx

...

13)3(

)3(3)2(8

17

k k

k

...

1

1

)!3(

)1(

32

3cos

9

2

9

1

k

k

k

ke

e

.163900632837673937287...

1 2

5

2

3log

4

1)2(

8

3log

kk k

k

.16395341373865284877... 3

1

12 2 1

41

e

e kk

2 .16395341373865284877...

0

2

)!2(1

1

2

1coth

k

k

k

B

e

e

4 .16395341373865284877... 3 1

1

12 2 1

40

e

e kk

.16402319032763527735... 1

12

2

3

4

3

3

4 9

1

6

1

31 1

21

( ) ( ) ( )

=k k k

kkk k

( )

( )

2

9

9

9 112 2

1

2 .16408863181124968788...

22 3

)(

k k

k

.1642135623730950488... 25

4

2 1

2 22

( )

( )!

k

k kk

!!

1 .16422971372530337364...

5

4

... loglog

224 4

2

2

1

2

1

2

2 2

2 2

Li

iLi

i

= log( )

( )

1

1

2

20

1

x

x xdx

1 .16429638328046168107...

13)!!(

1

k k

.164351151735278946663...

1

250 43

)1(

3

2log

331

xx

dx

kk

k

.164401953893165429653...

1

2

)1(2

)1(

2

3log

kk

kk

k

H

6 .16441400296897645025... 38

.16447904046849545588...

02

0

1

)2(!

)1(

)1()!1(

)1(1)1(

1

k

k

k

k

kkkk

kEi

e

= dxe

xxx

1

0

log

.16448105293002501181...

k

k

kk

H

kLi

2)1(2

1

2

122log

6

)2(

022

22

... 2

5160

11

logx

dx

x

...

1

0

24 log)1log()4(2

45 x

xx

.164707267714841884974...

13

42

)12(1288

2log)2log1(

8

)3(7

k

k

k

kH

.16482268215827724019...

primepk p

p

k

k

1

loglog

)3(

1

)3(

)3(3

23

... 1

4

.165122266041052985705... ( )( )

16 1

1

1

kk

k

k

... )5()3( 9 .1651513899116800132... 84 2 21

2 .16522134231822621051...

222

1)()1(

1

kk

k

kFkkk

k

= 2

5tan55

102

15

2

5

2

3

...

1

1

)!3(

)1(

2

3cos

3

2

3

11

k

k

ke

e

2 .1653822153269363594... e EiH

kk

k

( )!

11

... )1(

...

23

123

log

)!1(

1

log

1)3(1

k

n

nk k

k

nkk

k

... 2 3 4 2 3 4 24 1

2 2

1

log log log log( )

H

kk

kk

...

0 36

)1(

16

)1(32log

6

3

4 k

kk

kk

= dx

x x2 21 1

.165672052444307559...

0

logcossin5log2arctan210

1dxxxxe x

... 12

12

2

12

1

2

1

2

2 3

3

log logLi

=log2

3 21 2

x dx

x x

.165873416657810296642... e

e

8

)1cos1(sin1sin1cos1sinh1cos1sin1cosh

4

1 2

=

1

1

)!4(

4)1(

k

kk

k

k

.16590906347585196071...

123

2

53

1

250

63

50

3

3050

3log9

k kk

...

1

8

1

42

2

3 2 1 21

( )/ /

log x dx

x x

.16599938905440206094...

14

2

)2(1)4(4)3(4)2(

k k

k

...

1

2

)12(

1

!)!2(

!)!12(1

4

k kk

kG

J385

...

1

1

)!4(

4)1(1cosh1cos1

k

kk

k GR 1.413.2

... log3 2

2

... ( )

( )log2

3

23 2

1

4 2

2

4 31

k

k kk

= ( ) ( )k k

kk

2

22

.16658896190385933716... 2

2

.16662631182952538859...

4/

02cos)cos(sin

sin

8

2log

2

2log

4

dx

xxx

xx GR 3.811.5

.166666666666666666666 =1

6 1

11

4

32 2

12

3

k

k kk k k( )! k

=

1 4

1)2(

kk

k

=

123 )4)(3)(2(/1

1

kk kkk

k

kkk

=

0 )52)(12(

)1(

k

k

kk

=

2 1

1 2

)2(2

)1(

)2(!

1

k kk

k

k

k

kkk

=

32

41

k k

= k k

k k

k k

k kk k

( )

( )( )

( )

( )( )

4

2 2

6

5 11 1

J1061

= arccosh x

xdx

( )1 31

1 .166666666666666666666 =

1

4

)(2 2

)8(

)4(

6

7

k

k

k

HW Thm. 301

=

primep p

p4

4

1

1 Berndt Ch. 5

.16686510441795247511... sinh sin sin

( )

1 1

2 4

1

4

1

2

1

4 30 !

e

e kk

...

2 20

11

2

!!

1)(

k k kkI

kk

k

...

0

4/14/1

)!4(

4)1(cosh)1(cos2cosh2cos

2

1

k

k

k

...

0

2

!

)1(22

k

k

k

ke GR 1.212

... 8

arccos

2 .16736062588226195190... 22

2cos2cosh

=

142

4/34/1 11

)1(sin)1(sin

k k

.167407484127008886452...

1

2

1

6

)1(

13

13log32log

k

k

kk

...

1 )!1(2

11

k k

5 .16771278004997002925... 6

3, volume of the unit sphere in R6

...

loglog ( )

( )

22 1

22

kk

k

k

k Dingle 3.37

= 1

2

2 1

21 k

k

kk

log

.16791444558408471119... 9

2

3

2

1

3

12 2 1

91

cotkk

...

0 )!3(

1

2

3cos

2

3

1

k kee J803

...

1

3

)!3(

))!1((

k k

k

.16809262741929253132... ( )

( )

3 1

3

1 .168156588029466646... log

( )

2

2 2 1

k

k kk

...

1

4 1)3(k

k

...

0

12 )12(6

1

6

1arctanh

kk k

AS 4.5.64, J941

...

6

2

32

5 3

4

3 3 3

1

6

5 3

4

3 3 3

log

i

i

i

i

=1

2 1

3

831 1( )

( )

k

k

kk

k

...

1

1

23

1)13(2)1(2

2

k

kk

k

kk

k

...

144 6

1

6

1

kk k

Li

... dxx

xx

1

02

2

1

1log

322log2

2

3log3

=

12

2

1

1log dx

xx

x

.168547888329363395939... 4 16

1

8 2 1 2 2 1

2

4 21

( ) ( )k kk

.168655097090202230721... 4 2 2 14

2 2 1 8 27 3

16

2

log log( )

=1

8 2 1 2 2 14 31 ( ) ( )k kk

.16870806500348277326...

1

21

1

2 2

1

21

4 2

4

2 1

81

( )kk

k

= 1

8 31 k kk

...

2

1)(

kkk

k

...

5/4

1

5/1

1

...

23 3

)1(

k

k

k

.16945266811965260313... 14

27 92

0

1

x x xarcsin arccos dx

...

32

3cos

2

9

1 e

e

.16948029848741747267... ( )k

kk

4

42

= 1

1

1 14

24 4

212 4

220 4

2k k k k k k

1

kk k k k

= ( ) ( ) ( ) ( )4 1 8 4 1 16 4 1 16 4 11 1 1 1

k k k kk k k k

.16951227828754808073... 21

2

12 2 1

41

cotkk

=

112

1tan

2

1

kkk

Berndt ch. 31

... 9log2

11 .1699273161019477314... 12 22 6 2 23 2

0

x xd

ex

log x

...

0 )12(6

12

2

3arcsin

2

3

kk kk

k

... k

k

kk

1log

1

22

...

133 6

1

6

1

kk k

Li

...

1

0

2

2

1

1)1log(

121

xe

dxx

eLie

...

2

1)(k

k

... 4/34/34/5

2coth2cot214

23

=

12

41)4(8

8

8

k

k

k

kk

...

k

k

k

k

k 9

)2()1(

19

1

2

1

3coth

61

12

...

12

)1(

)1(3

16

3

2

9

1

12

5

362

3log

k kk

k

=

1 3

)1)((

kk

kk

... kkF

8

3

.1710822479183635679... ( )k

kk 41

...

1)(

1)13(

8

1

k

k

... Re ( ) ( ) i i i

...

0 )!2(3

1

3

1cosh

kk k

.17138335479471051464... 41 1)1()1( kk

...

12

1 1( ) ( )k kk

1 .1714235822309350626...

1

428

)()4(8100 k

kkd Titchmarsh 1.2.1

...

1

2

5

41

21

log coscosk

kkk

.1715728752538099024... 3 2 22 1

8 1

2 1

2 2 11 1

k

k k

k

k kkk

kk( )

( )!!

( )! ( )

=( ) ( )!!

( )! ( )

( )

( )

1 2 1

2 1

2 1

4 1

1 1

11

k k

kkk

k

k k

k

k k

=

0

)2()!2(

)1(

2

1

k

k

kk

... ( )

( )

2 1

1

112

1

22 2

2

k

kk Li

kk k

.17172241473708489791... 7

)3(

... 6 3 1 5 13

1

cos sinlog coslogx x

xdx

e

.17186598552400983788... 11

22 2( ) ( )i i

= 1)1()1(2

11)()(

2

1 iiii

= 11 2 13

2

1

1k kk

k

k

k

( ) ( ) 1

=

1

)14()14(k

kk

= 1

10

cosx

e edxx x

.17190806304093687756... dx

x x x4 21

...

i x

e ex x

3

42 1 1

12 1 3 2 2 3

0

( ) / ( ) /( ) ( )dx

GR 3.418.1

= 1

3

1

3

2

3 11

2

20

1

( ) log

x

x xdx GR 4.23.2

5 .172029132381426797... dxx

x

1

03

32

)1(

log

4

)3(9

1 .1720419066693079021...

1

3)1(4

11

2sinh

5

8

k k

...

22 2

log

k k

k

...

1

2/1)1(4

tanh2

1

2

1

k

kk e J944

... 2e

...

1

2

)!3(k k

k

.17229635526084760319... ( ) ( )

( )

( )2 3

4

4 1 31

H

k kk

k

... 1)2(49

4

16 1

2

kk

kk

.17250070367941164573... 11

2

1

2

12 2 1

21

cotkk

... dxxx

li )log3sin(1

23log3

10

1 1

0

GR 6.213.1

.17254456941296968757...

1

1

223

3

1)3()1()1( k

k

k

kkk

k

...

13

1

k kF

...

1

2

)2(32

2

3log4

2

3log3

2

3

kk

k

k

H

.17260374626909167851...

1

21

12 2)!2(

4)1()2(1sinlog

k

kk

k

kk kk

B

k

k

AS 4.3.71, Berndt Ch. 5

... ( ( ))2

...

12

23

11)()1(

kk

k

kkLi

k

k

...

1

1

)3(!

)1(

3

55

k

k

kke

.1728274509745820502... log ( )2

2

1

2 1

1

0

GH

k

kk

k

Adamchik (24)

= dxx

x

1

02

2

1

)1log( GR 4.295.5

...

2 2

log

kk k

k

1 .17303552576131315974...

1 2

cot22

11)2()2(

k k kkkk

...

1

1

)!1(

)1()1(2)1(

2

k

kk

k

kH

e

EiEi

e

...

n

mn

nmhn 1

)(1

lim)3(

)2/3(

h(m) = lowest exponsent in prime factorization of m.

... 11

22 2 e ei i

k

= ( ) coskk

1 11

.17328679513998632735...

1 )14)(24)(34(

1

4

2log

k kkk J253

=

0 44

)1(

k

k

k

=

1 12)14(

1

r srs J1124

= 1

8 2 1 2 2 131 ( ) ( )k kk


=

15 xx

dx

= dxx

xx

1

022 )1(

log GR 4.234.2

...

12!!

1

k kk

...

2 !

1)(

k kk

k

.17352520779833572367...

03 32

)1(

k

k

k

...

1 !

)1()1(

k k

keEi

... kk)6(

1

...

14)1(

)5(2)3()2(4

)4(

k

k

k

kH

...

1 )13(3

1

kkk k

.17417956274165000788...

1

1

122 5

)1(

6

1

6

1

kk

kk

kk k

H

kLi

.17426680583299550083... arcsinh11

2 1 1

2

2 20

1

x dx

x x( )

...

2

3 216

3 3 9 3 6 4 361

6

log logk kk

= ( )( )

12

61

1

kk

k

k

2 .17454635174140...

1

/11)1(

k

kk

k

e

...

12 2/1

11

k k

...

1

41)(k

k

... )3(log)(1

13

kk

k

p kprimep

.1750000000000000000 = 40

7

1 .1752011936438014569... iiee

sin2

1sinh

2

1cosh2

21sinh

1

=1

2 10 ( )k !k

AS 4.5.62, GR 1.411.2

= 112 2

1

kk

GR 1.431.2

= cosh1

12x

dx

x

.17542341873682316960... cosx

edxx

20

.175497994056966649393...

1 )!1(

log

k k

k

... 1

121 e

k

k

.175639364649... 1

221 n

k

k

n

kkn !

log

!

.17575073121372975946... 83

8 20

1

e

x x

edxx

sinh

.175781250000000000000 = 45

256 9

2

1

kk

k

.17592265828206878168... eFkk

e

k

kk

;3;2

1

)!2(!

)1(10

0

.17593361191311075521...

22 2

11log

2

1

)1(2

1)(

kkk kkk

k

1 .1759460993017423069... i i i x

xdx

4

1

2

1

21 1

0

( ) ( ) sin

sinh

x

.17606065374881092046...

1

1

13 6

)3()1(

16

1

kk

k

k

k

k

.176081477370773117343...

1 3)!2(3

1sinh

32

1

kkk

k

... 4 3

20

3

08 1

log

sinh

x dx

x

x dx

x

1 .17624173838258275887... )(

1 .176270818259917... smallest known Salem number

.176394350735484192865... 20

1

20

2

10 61

log arctan x

xdx

1 .176434685750034331624...

12

1

2

)1(1

k

k

k

.176522118686148379106... 3

2 2

8

21

1 1

21

1

log( )

( )k

k

k

k

= 1

21

11

2 kk

kk

log

.176528539860746230765... 12log2/11

2/11log

24

3

=

0 )3)(2)(1(2

1

kk kkk

J279

.17659040885772532883...

1

1

13

)1(

kk

k

.1767766952966368811... 1

4 2

1

4

2

0

( )k

kk

k k

k

... /1e

.17709857491700906705... 5 1 3 11

2 1 2 31

cos sin( )

( )!(

)

k

k k k

=log sin log

sin3

15

1

1x x

xdx

x

dx

x

e

1 .177286727908419879284... 1

2

2 1)(

k

k

k

.17729989403903630843...

0

1

)53)(13(

)1(

8

1

36 k

k

kk

1 .17743789377685370624...

0102 2)!2(!

12

2

1;3;)2(4

kkkk

FI

.177532966575886781764...

12

22

2

)2(

1)1(

121

kk

k

kkk

=

1

22

22

2

2cos)1(

2

1

k

kii

k

keLieLi

=

02

12

1

0 )1(

loglog)1log(

xx ee

dxxdx

xx

xdxxxx GR 3.411.10

= x

e edxx x

10

=

x x

xdx

log

10

1

=

1

0

1

0 1dydx

xy

yx

.17756041552698560584... 1

82 3 2 3 3

2 11

log( )cos

k

kk

J513

2 .1775860903036021305... 2log

= dxx

x

1

121

)1log(

=

02

2

02

2

4

)4log(

1

)1log(dx

x

xdx

x

x

= dxx

x

02

2

1

)1log( GR 4.295.15

= 2

0

2

1

1log

x

dx

x

x

GR 4.298.9

=

0

tan dxxx

= 2/

02

2

sin

dxx

x GR 3.837.1

=

0

2

1

1

arctanarcsindx

x

xdx

x

x

=

2/

00

)tan4log(2

sinlog

dxxdxx

= 4)3(

1728

2035H

1 .17766403002319739668... 7

...

012 )12(2)!12(

12log

2

1

kk kk

ci

...

1

3

)!3(k k

k

.177860854725449691202...

2

3

3 6

2 3

1( ) ( )

( )

n , where n has an odd number of prime factors


... 4/

02

22

cos

tan

4162

2log dx

x

xx GR 3.839.3

...

12 1

1

kke

Berndt 6.14.4

... !4log

1 .17809724509617246442... 21arctan8

3

= ( )

( )sin

1

2 1

6 3

2

1

21

k

k k

k J523

= ( )

( )( )

k

k k

k k

k kk k

1 4

4 8

2

12

321

2

21

8 4

3

= dx

x( )2 30 1

= x

xdx

3 2

0

1

1

/

GR 3.226.2

= sin3

0

x

xdx

GR 3.827.4

.1781397802902698674... 12 24

2 2

20

log sinx x

xdx

...

126k

kk

k

H

... ( ) ( )

( )(

2

2 4

1

2

1

1 21

)

k

k k kk

... cosh( )

22

2

2

2 2

0

!

e e

k

k

k

GR 1.411.2

... k

kk

2

2 !!

...

1

2

!!2

2

1

233

k k

kerf

ee

.1782494456033578066... 1

2 42 4 2

1

2

arctan

dx

x x

...

0

4

242

)22(

)12(

4

)3(

14424 k k

k

... k k k

kk

! ! !

( )!31

... 1

162

12 1 1

2

1

log ( )k

kk

k

...

0 )5)(2(!

13

3

25

k kkke

.1785148410513678046... 4

5

5

41

41

1

log ( )

k kk

k

H

.1787967688915270398...

4 3

3

4

1

3 3 1 3 21

log

( )(k k k )k

J250

= dx

x x4 21 1

... 21

21

1

2

20

1

arctantan sin

cos

x

xdx

... 2

5 5

1 21

21

( ) ( )!

( !)

k

k

k k

k

2 .17894802393254433150...

2

21)(2k

k k

.178953692032834648497... 1

2

1

2

1

12 21

cot

kk GKP eq. 6.88

=

1

2121

12 )!2(

2)1()2(

k

kkk

kk k

Bk

... 8

1 11

41

3erfi erfx

dx

x

sinh

... 5 2 61

1 2

1 12

1

( ) ( )!

( )!

k

kk

k

k

...

0 )4(2

)1(

2

3log16

3

20

kk

k

k

.1792683857302641886... 2 3

4

1

2

1

41

2

20

1

log

log( )e e

dx

e ex x

.17928754997141122017...

2 1)(

1)()1(

k

k

k

k

.179374078734017181962... ee

ke

k k

k

11

1

1

1

( )

( )!

.17938430648086754802... 1 1 11

11

( ) log( ) log( )

k

k k k J149

376 .179434614259650062029... 145739620510

387420489 10

10

1

kk

k

.1795075337635633909... 2 2 2 2

204 2 24

2

2

2

2

2

4

log log log logx xd

e x

x

5 .179610631848751409867... e

1 .17963464938133757842... 1

6 4 3

3

2

1

4 321

cot

kk

... )32log(2

3

4

3

3

1arctanh

2

3

4

3

=

0 )12(3kk k

k

... 3

1

35

6

2 22 1

2 31

log

( )!!

( )! ( )

k

k k kk 1

... 7

6

7

6 71

log

Hkk

k

.180000000000000000000 = 50

9

... 5 3

16

2

0

1 ( ) arcsin arccos

x x

xdx

... ( )

! (

1

4 2 1

1

1

k

kk k k )

Titchmarsh 14.32.3

...

0

2

2 32

12

4

3 kkk

k

.1804080208620997292... 23 1

1 20

e

k

k

k

kk

( )

( )!

.18064575946380647538... 50

10 5 51

201 5 2 5 2 ( ) arccsch arcsinh

=

log5 1

32

= ( )

1

5 1

1

1

k

k k

5 .180668317897115748417... )2(e

.18067126259065494279...

1

222

22

)58(

1

)38(

1

8

3csc

642216 k kk

= dxxx

xx

1

042

2

)1)(1(

log GR 4.234.5, Prud. 2.6.8.7

.1809494085014393275...

8 3

3

24 831

log dx

x

... cosh

sinh1 1

3

13

14

x

dx

x

...

2 12

21

kk

...

21 2

11log2

1

)1(2

1)1(

kkk kkk

k

3 .18127370927465812677...

1

1

2

)!12(

1)2(2

k k

k

kI

.18132295573711532536... 6)1(

1 .18136041286564598031... e1 6/

1 .1813918433423787507... dx

x!1

1 .18143677605944287322...

012

3/1

)13(2

11,

3

4,

3

1,

3

12

kk k

F

...

1

1

!k

kk

k

FF

1 ...

primep

p 31 Berndt 5.28

=

1

3

)(

)6(

)3(

k k

k

Titchmarsh 1.2.7

=

squarefreeq

q 3

...

0

4

22 )(sin

3

2dx

x

x GR 3.852.3

.18174808646696599422... 2 2

124 2 1 2 2

2

24 arcsinh log

log

=1

2 2 121

kk k k( )

...

3

5

2

3log3

322

3

... 5

cot2

25)3(10log25

6

5125

2

5

2sinlog

5

4cos

5sinlog

5

2cos50

=

k

k

k

k

kk 5

)3()1(

5

1 1

134

...

123

2

34

1

27

16

9

2

4

2log3

18 k kk

= 11

2

... 612

6 2 6 33 1

3 1

1

6

2

3 21

log log( )k

k k k

... ( )k

kk 31

... HypPFQk

kk

1

211

1

4

3

4

1

16

14

20

, , , , ,

.18214511576348082181... 2 2

1

1122

2

21

1

loglog

( )( )k

k

k

k

...

212 3

)3(

)13(3

1

13

1

kk

k

kkk

kk

.182321556793954626212...

11 6

10,1,

6

1

6

1

5

6log

kk k

Li

=

0

12 )12(11

1211arctanh2

kk k

J248

.182377638546170138737... 1

42 2

3

2

1

8

12

22 1

1

1

Fk

k

k

k

kk

, , ,( )

... 11

6 4

2

23 1 3 2

1

32

log( ) ( )

( )k

kk

... ( )

1

3

1

31

k

k k

... ( ) ( ) ( )( ) ( )3 11

42 22 2 i i

=x x

e edxx x

2 2

0 1

cos

( )

...

14

12

2 log

)3(

1

9

)2(

54 xx

x

kk

... e10

.18316463254842946724... 1

41 2 1

2

1

log( cos )sin cos

k k

kk

... 5

2 22 2

2

27 41 4 1 4

42

/

/ /cot cothkk

= 2 4 11

k

k

k ( )

... ( ) ( )

arctan

1 1

2 1

1 1 1

2 2

k

k k

k

k k k

k

.183392750954659037507...

0

)1(

22

1

2

1

k

k

k

...

2

1

2

1

4

1

2

3

4

1

2 10

, ,( )

k

k k

= dx

xlogcot

/

0

4

GR 3.511.

... G

2

.1835034190722739673... 12

3

1 2 1

2 2

1

1

( ) ( )!!

( )!

k

kk

k

k

... dxee

xxx

0 1

sin1

2

coth

.183578490978106856292...

12

)()1(

2

1

14

)2(

12

)2(

11k

k

kk

kk

kkk

.18359374976716935640... ( )k

k

k 221

...

...

11 14

)2(

12

)2(

kk

kk

kk

...

3

1)3(

... e e 2cosh

...

1 )!12(2

1

k k

k

e

=

1

2xe

dxx

= 3

15

12

1sinh

2

11sinh

x

dx

xx

dx

x

...

12 644

1

6

1

2

5tanh

20

5

k kk

=

1

2

4125

148

kk

k kF

... 1

3 2 1 22 k kk

/ /

... 1

3 22 k kk

/

...

primep p 31


=

23 log

)(

k kk

k Titchmarsh 1.1.9

3 .18416722563191043326...

121k k

k

...

1

16 443

5log

xx

dx

... 3

4

3 3

42

6 1

12

121

log ( )!

( )!( )

k

k kk

.18432034259551909517...

012

1

)4)(1(

)1(

3

)1(

18

5

3

2log2

k

kk

kkkk

= 4

1

1

0

2 11log)1log(

x

dx

xdxxx

... 3

32

20

27

2 1

4

4 4 1

4 1

2 2

1

2 2

2 32

k k k k

kkk k

( ( ) ) ( )

( )

...

00

4/1

2!)!12(

1

)!12(

!

2

1

kk

k kk

kerfe

= dxe

ex

x

1

0

2

...

21

121 2sin

1

)!12(

)1)2((2)1(

kk

kk

kkk

k

... )6()3( 1 .184860269128474... 1 6( )

.184904842471782293011...

2

/1

1

12

11

!

)1)12(()1(

k

k

k

k

ekk

k

...

1 1

2

2

/11

1

kxk

dxe

x

e

k

eee

... 8

491

8

7 81

logkHk

kk

... 1 3 2( / )

... sin1

31

3x

dx

x

8 .1853527718724499700... 67

...

1 !

)(1k

kk

e

k

HeeEie

...

1

164

2

48

2

12

1

16

1

43

2 3

3Li Li( )log log

=log2

31 4

x

x x

...

111112 2

)(

4

1

22

1

14

2

12

1

kk

kkk

kk

k

kk

k

... 256103

2610003

5 H ( )

...

32 22 2 2

2 12

2

2 21

csc sin( )

k

kk

... 4/

0

tan8

2log

2

dxxx

G GR 3.747.6

...

22

1

1 11log

11)12()1(

kk

k

kkk

k

... 1

22 1 3 2 2

221 1

1

log log( )

Hk

k

...

2

/3

2

1!

)1)((3

k

k

k

k

ek

k

.18618296110159529376... ( )

1

2 20

k

kk

... 23

3

12

2 11

k

kk k

k( )

= dxx

x

1

031

1log

...

0

4

3 )!34(2

sinh

k

k

k

... H

kk

kk

( )2

1 6

... 8

6 21

2 4

2 3

1 3log log

arcsinh

= arcsinxx

xdx

1 2 20

1

GR 4.521.3

.18646310636181362051...

( )3

41

kk

k

...

1

12

2

!

2)1(

)2(2

k

k

k

k

k

k

e

I

.18650334233862388596... 1

11 33

2

1

1kk

k

k

k

( ) ( ) 1

=

3

5

6

1

61 3

1 3

21 3

1 3

2

i

ii

i

1 .18650334233862388596... 1

11 1 35 4 3 2

0

1

1k k k k kk

k

k

k

( ) ( ) 1

=

23 12

1

k k

k

=

2

3131

2

3131

6

1

23

ii

ii

= 1

63 2 2 1 1 2 1 2 1 2 12 3 1 3 2 3 2 3 ( ) ( ) ( ) ( )/ / / /

1 .1866007335148928206... k

ekk

11

.1866931403397504774... 1

21

1

23 4 1

1

2

21 1

1

1

F ( , , ) ( )( )!

k

k k

k

k

... 2

12sin

4)!(

)!2()1(

4

1sin

)2/1cos(2

1

02

k

k

k

kk

k

...

...

1156

2log

36 xx

dx

...

12 2

1

2

7tanh

7 k kk

1 .18683727525826858588...

0

3

3)!1(

1133

kkk

...

1

3

12

)23(

)1(2736)3(6

16

1

k

k

kk

... k

k

k k

k

k

3

2)1(

7

3

7

2

0

...

2 )1)((2

1)1(

kk k

k

...

1 23

1

32

1arcsin11

121

12

11

1

k k

k

k

... 7

K Ex. 108d

.187500000000000000000 =

1

1

3)1(

16

3

kk

k k

=

21

1

)2)(2(

)1(

kk

k

kk GR 0.237.5

=

1 116

4)(

kk

kk

... 12 24

22

2

22

0

1

i i i

Li e x x dxilog ( ) log( cos )

.188016647161420393823...

2 )1)(2(

1)(

k kk

k

... 9

1

... dxx

xG

1

21

)1log(

8

2log

= GR 4.227.13 4/

0

)cot1log(

dxx

...

1/1

2

11

)!1(

)()1(

kk

k

k

kekk

k

.1883873799298847433... 21log22

1

2

1

= 1

42 2 1

1

21

1

2

1

2

arcsinh arctanh

= 1

2 2 1 2 11k

k k k( )( )

1 .18839510577812121626...

0

2

)!2(

42)1(1csc

k

kkk

k

B [Ramanujan] Berndt Ch. 5

...

12

2

)2)(4(6

11

k kk

k

...

24 10

1

k k

.18872300160567799280... 2

)21(82log4)21log(2

=

8

3sinlog

8sinlog22log4

8cot

28

=

k

k

k

k

kk 8

)1()1(

8

1 1

12

Prud. 5.1.5.21

= dxx 1

0

6 )1log(

... 4

91

16

9

1

41

log( )k Hk

kk

.1887703343907271768... 1

2 1( ) e J147

... 3

4 , volume of the unit sphere

3 .1889178875489624223... sinh

2

3 21

22

2

kk

...

12 )!2(!

1

2

1)2(

k kkI

...

1 )1(3

1

3

2log21

kk kk

J149

=

1

1

)1(2

)1(

kk

k

k

... 2 11

4 34

0

( )k

k k

... )1()1()()( iiii ... )2()2( ii

... 1 4

32

0

1

x xsin dx

...

12

/1

2

12

)!2(

)()1(

k

k

k

k

kke

k

k

... dxexe

erf x

1

2 2

2

1

4

1

...

1

1 2

)12(2

)1(

2

2arcsinh21

kk

k

k

k

k

...

2

22 1)(k

kk

... dxxx

xdx

xx

x

2

12

21

0

23 log

)1(

)1(log

3

2log

4

)3(

.189727372555663311541...

122

1 )18(

8

4

)12(

2

11

22

11

28

1

kkk k

kkk

406 .189869040564760332...

0

2

!

cosh

24

1 22

k

ee

k

keee

.1898789722210629262... 1

2

1

1

1

21

csch

2

( )k

k k

...

2 5

517

20

1

5

1

23

csc( )

k

k k

...

12

1 9

11log

9

)2(

33

2log

kkk kk

k

...

22

1 11log

1)1()(

)1(

kk

k

kk

kkk

k

... 13

13log

... 1

1 32

kk

1 .190291666666666666666 = 6)3(

24000

28567H

...

1

21

21

2)!1()!1(

)!()!(124

2

7

kkkk

kkE

.190598923241496942... 15 8 3

6

1

6 2

2

1

kk k

k

k( )

...

2

1)(12

)1(

kk

k

k

...

3

7

...

12 1

2 1

2

20

1

arctan tan sin

sin

x

xdx

... 11

220

( )k

kk

.1906692472681567121...

1 )!2(

1

2

)1(

24

9

k kk

Eie

=1

1 33 ( )! !k kk

...

12

22

66

126log6log5log25log

2

1

kk

k

k

HLi

=

12

1

5

)1(

kk

k

k

... 53

...

22

22

111)(

kk kkLi

k

k

...

12 544

1

5

1

8

tanh

k kk

...

12

1

2

)1(

k

k

k

... 2

8

.19166717873220686446...

( )2

51

kk

k

.19178804830118728496... 4 2

3

2 3

3 41

2log log

dx

x x

... ( )

( )

2 1

3 1

...

11

1 1sin

1

)!12(

)12()1(

kk

k

kkk

k

... G2

... 1ee

e J152

... log k

ekk

1

.1923076923076923076 =

0

5sin

26

5dx

e

xx

=

0

)12/()5(log)1(5logcosh2

1

k

kk e J943

.192315516821184589663... 3

... 3

...

1

3

8kkkH

.19245008972987525484...

0

2

2

)1(

33

1

kk

k

k

kk

.19247498022682262243... 2 6 2 3 62 1

2 31

log log( )

( )!

k

k kkk

!!

... 26

3

...

1 2 11 2

0

e Eix

e xdxx( )

( )

...

4

3

4 4

1 2

2 1

1

0

loglog ( ) log( )

k

k

k

k

1


...

129

11

3sinh

3

k k

.193147180559945309417...

222 )12(2

1

24

1

2

12log

kk kkkk

=

1 )12(2)12(

1

k kkk J236

=1

8 231 k kk

[Ramanujan] Berndt Ch2, 0.1

=

15

1

0 4

)1(

3

)1(

k

k

k

k

kkk K Ex. 107f

=

2 2

1

kk k

=

0 )3)(2)(1(2

1

kk kkk

J279

=

112

2 2

)12(

2

1)(

kk

kk

kk

=

=

134

223 xx

dx

xx

dx

=dx

x x x

dx

x( )log

11

12

12

13

=

02 sinh)1( xx

dxx

GR 3.522

= dx

e ex x20 1

... 7)3(

8232000

9822481H

... 2

9

32

2

21

k

k k

kk

... e i i / /4 2

...

1

2/1

1

1)12(!

1

kkk

k

ek

...

1

1

)!1(k

ke

k

ee

... )7()3(

... 1log1

!

1)3()1(

3

21

1

k

kk

k ekk

k

...

log log log2

1 3

2

1 3

2

i i

=

3/23/1 )1()1(2

1log

= log ( )( )

11

13 1

32

1

1

k

k

kk

k

k

.194035667163966533285...

0

2 32

)1(

2

3csch

3

6

6

1

k

k

k

... ( ) ( )

log

1 1

1

1

21

1 1

2

1

422

2

2

k

k k

k

k

k

k k

k

...

1

)3(1

4)1(

kk

kk H

... 2

1)1(1)1(1

44/14/1 i

4/34/3 )1(1)1(14

i

=

1

1

24

1)14()1(1 k

k

k

kk

k

...

122 2

1

22

2cot

4

1

k k

.194444444444444444444 =

11 )3)(1(

10,1,

7

1

736

7

kkk kkk

k

... 2

1

... e k

k

k

k

2

1

3

2

2

2

( )!

... e

k

k

k

2

0

1

2

2

1

( )!

... e k

k

k

k

2

1

1

2

2

1

( )!

... 2 2 22 1

12

0 11

e I Ik

k kk

( ) ( )( )

!

.194700195767851217061... 1

41

41 41

1e

k

kk

kk

/ ( )!

... 9

... 2

1

2

1

2 2 10

erfik kk

k

! ( )

... 16

sin

... 4/

04

3

cos

sin

3

22

dxx

x

...

12

24

2

3

)12()2(

3

11

3cot

3)2(

kk

k

kk

kk

k

...

1

1)4(2

k

kk

k

...

1 )!1(k

k

k

kB

... dxe

xHypPFQ

x

1

222 log

1},2,2,2{},1,1,1{26

... 2

cot2

...

1

123

3

72

3

72

1

12

1

33

2 3

3Li Li( )log log

=log2

31 3

x

x xdx

....

2

2)1)()1(

k

k

k

k

... ek

k kk

(cos )/ cossin cos

!1 2

0

1

2 2

GR 1.463.1

= 2/2/

2

1 ii ee ee

...

0

2

6

1133

kk k

k

5 .19615242270663188058... 27

....

1

2

1

13

)1(

3csch

322

1

k

k

k

...

1

2

92

2log3

8

3log9

kk

kH

...

21

2

0

tanh sinsinh

x dx

x

.19634954084936207740...

16

=

04 4)12()12(

)1(

k

k

kk K Ex. 107e

= ( )

( )sin

1

2 1

2 1

4

1

21

k

k k

k J523

= sin cos sin cos3

1

3 2

1

k k

k

k k

kk k

=

22

2

032

2

)4()1( x

dxx

x

dxx

=

044 xx ee

dx

1 .1964255215679256229... H kkk

21

2 1( ( ) )

... 44

2 2

20

1

Gx dx

x arcsin

= dxx

xx

2/

02

2

sin

cos

GR 3.837.5

= dxxx

x

022

3

1

arctan GR 4.534

... 14},1,1{,2

12

)!(

1

12

HypPFQk

k

kk

...

3

1

6

3log

318327

2

9

)3(13 )1(3

=

12)13(k

k

k

H

...

0

2

10

!

2)1(22

k

kk

kJ

... e

e

k

e

k

kk( )

( )

1

12

1

1

.1967948799868928... ( )( )

15 1

1

1

kk

k

k

... 2

3

... 23 3

2 1

2

0

( )

)

x

e edxx x

=

023

2

123 xx ee

dxx

xx

dx

... 9

... ( )

! ( )

1

2

k

k k k

... ( )

! ( )

1

2

k

k k k

... 25 1

32

27 1

32

13

16

1

32 2

5

1

cosh sinh

( )

e

e

k

kk !

... 3log2

... 12

2

1 4

2 1 2 1

1

1

si

k k

k k

k

( ) ( )

( )!( )

...

12

)1(2

)4/1(

1

4

18

k kG

.19739555984988075837...

0

12 )12(5

)1(

5

1arctan

kk

k

k

=

1

2)2(2

1arctan

k k [Ramanujan] Berndt Ch. 2, Eq. 7.6

...

0

12 )12(5

)1(

5

1arctan

kk

k

k

...

1

2

)2)12()2((128

3

k

kkk

.19749121692001552262... 2 13 1

21

2 11

2

1

cossin

( )( )!( )

k

k

k

k k

...

12 116

21

4coth

4 k k

...

11 9

)2(

)13)(13(

1

362

1

kk

k

k

kk

... 1

254 3 2 2 5 4 2 1

41 2

1

arccot log log ( )k kk

k

kH

... 60 222 101

ek

k kk !( )

.19783868142621756223... 2 2 2

04

2

2

2

4 48

log log log sinx x

xdx

2

... 19

12 327

13 3

8

1

16

7

6

1

1 3

32

1

31

( ) ( )

( /( )

k

k k )

.19809855862344313982... 11 5 23 2

2

1

1k kk

k

k

k

( ) ( ) 1

...

( )log

33

1

k

kk

...

4

54

3

2)1()1(2

KE

= dxxdxx

x

2/

002

2

sinsin1

sin

...

1

2)12()2(k

kk

...

0

21 )32()!)!12((

)1(2)1(

k

k

kkH

...

1

21

2

)1)1(()1(

kk

k k

...

1

4

)!1(

4

4

13

k

k

k

ke

...

0 23

)1(

kk

k

...

4 32 3

17

66

1

1224

csc( )k

k k

... ( )

! !!

1

0

k

k k k

... 1

0

2 log)1(2223 dxxxeEie x

... ( )( )

log ( )

1 21

1

k

k

k

kk

1

...

1 2log

k

k

kk

...

1 )74(

1

7

8log

14147

40

k kk

... 8

1

...

3

2

3

13log3log2log

6 222

2

LiLi

... 2

22

1

2

2

0sintan

kk

k

k

Berndt ch. 31

... 4

21

22 2

1

21 1cos ( ) sin ( ) cos CosIntegral SinIntegral

= dx

e xx 20 4

...

13

1

2

)1(

k

k

k

k

... root of ( ) ( )x e x ex x 1 1

... 3

.20000000000000000000 = 1

5

1

6 5 1

2

1 1 0

4

1

kk

kk

x

ek x dx

e

x dx

x

( ) cos log

= k k

k kk

( )

( )( )

5

4 11

J1061

.20007862897291959697... k

kk

k

k

k

3

62

1

111 6 3

1

( ) ( ( ) )

=

2

34)31()2()2(

6

1 iiii

2

34)31(

2

34)31(

ii

ii

2

34)31(

ii

.2000937253541084694... 3

2 2 31

3

1

21

log( )k

k k k

=2

13

1

0

3 11log)1log(

x

dx

xdxx

.2000937253541084694... 3

2 2 11

30

1

log log

xdx

.2002462199990232558... 2 2 2 1 2 21

2 2 11

log( ) log( )k

k k k

= 2 2 2 1 2 arcsinh log

.2003428171932657142... ( )3

6

1 .2004217548761414261... 1 1 1

¼ ¾ ¼ ! ¾ !

.2006138946204895637... 1 2

2

1

81

21

2

3

2 2

3

2 2

log ii

i i ii

i

= 1

8

1

2 2

1

2 21

21

2 2

i i i itanh

3

4

2

2 8 2

logcoth

= ( )

1

1

1

3 21

k

k k k k

.2006514648222101678... Ei Ei( ) ( ) e e e x

10

1

dx

...

0

2

)1()!2(

4)1(1sin2sin

k

kk

kk

6 .201255336059964035095... 3

5

.2013679876336687216... 9

8

2

3

2

0

e k

k

k

k ( )!

.20143689688535348132... ( )( ( ) )

!

11 2

2

k

k

k

k

.2015139603997051899... 1

6

6

3

3

2

1

2 320

csch ( )k

k k

1 .20172660598984245326... 1

3 21k

k

.20183043811808978382... e

e

k

kk8

3

8 2

2

1

1

( )!

... e

e e

k

ek

kk

sin

cos( )

sin1

1 2 112

1

1

.20196906525816894893... 16 4 3 5 8 12

12

( ) ( )

i i

=1

41

2 5

47 51

1

1k k

k

k

kk

k

( )( )

... 1

2 1G

.2020569031595942854... ( )log

3 11 1

2 132

2

0

1

k

x x

xdx

k

GR 4.26.12

= 6 2

1 1

3

2 30

2

x x

x

dx

e x

( ) Henrici, p. 274

=

1

0

1

0

1

0 1dzdydx

xyz

xyz

1 .2020569031595942854...

1

32

2

13 )1(

1331)3(

kk kk

kk

k

= )1,2(MHS

= H

kk

k ( + )1 21

Berndt 9.9.4

=

122 )1(

)12(

k

k

kk

kH

=

1

)2(

)1(k

k

kk

H

= 5

2

12

5

2

1

2

1

31

1 2

31

( ) ( ) ( !)

( )!

k

k

k

kk

kk

k

k k Croft/Guy F17

Originally Apéry, Asterisque, Soc. Math. de France 61 (1979) 11-13

= ( )( !) ( )

(( )!)

1205 250 77

64 2 11

10 2

5k k k k

k Elec. J. Comb. 4(2) (1997)

=

1

23

3

123

3 )(2

180

7

1

12

180

7

kk

kk e

k

ek

Grosswald

Nachr. der Akad. Wiss. Göttingen, Math. Phys. Kl. II (1970) 9-13

=

2

12

)(4

45

2 22

123

3

kke

k

kk

Terras

Acta Arith. XXIX (1976) 181-189

= 4

7

2

4 2 2

2

72

7

2

0

2 2 ( )log

k

kkk

Yue 1993

= 4

9

2

4 2 3

2

92

2

27

2

0

2 2 ( )log

k

kkk

Yue 1993

=

22

4 2 2 2 32

0

( )

)(

k

k kkk

Yue 1993

=

4

7

2

4 2 1 2 2

2

0

( )

( )(

k

k kkk )

Ewell, AMM 97 (1990) 209-210

=

8

9

2

4 2 1 2 3

2

0

( )

( )(

k

k kkk )

Yue 1993

=

0

2

)32)(22)(12(4

)2()52(

3 kk kkk

kk Cvijović 1997

=

0

22

)32)(22)(12(4

))2(1)(52(

32

3log31

3

4

kk kkk

kk

Cvijović 1997

=

0

122

)22)(12(16

)2()12(4

4log

2

3

14 kk

k

kkk

k

Ewell, Rocky Mtn. J. Math. 25, 3(1995) 1003-1012

= primep

p 41)6(

=

primep pp

ppp21

321

1

1)4(

= 5

41 1 1 1

3

22 2

1

4HypPFQ[{ , , , },{ , , }, ]

= dxx

x

1

0

2

1

log

2

1 GR 4.26.12

= log( ) log10

1

x xdx

x GR 4.315.3

= 4

32

0

4

log(cos ) tan/

x x d

x GR 4.391.4

= 16

32

0

1

arctan log (log )x x x d x GR 4.594

= x dx

ex

5

0

2

1

= 2

3 1

3e

edx

x

e x

GR 3.333.2

= 1

2 1

3e

edx

x

e x

GR 3.333.1

2 .2020569031595942854... ( ) ( ( ) ( ))( )

3 1 22 1

12

2

32

k k kk k

k kk k

... )1)1316((1

12133

kk

kkk

... )1)1215((1

12143

kk

kkk

.2020626180169407781... ( )log ( )

12

k

k

k

k

... )1)1114((1

12113

kk

kkk

... )1)1013((1

12103

kk

kkk

... )1)912((1

1293

kk

kkk

.20211818111271553567... )1)811((1

1283

kk

kkk

.20217973584362803956... )1)710((1

1273

kk

kkk

.20218183904523934454... ( )( )

!

12

12

2

2

1

kk

k

k

k

k

ke

k

.20230346757903910324... )1)69((1

1263

kk

kkk

.2025530074563600768... 4/34/1 )1(cot)1(cot82416

7

4

ii

4/34/1 )1()1(4

)2()2(8

1 i

ii

= 1

8 5 13 5

2 1k kk

k k

( ( ) )

.2026055828608337918... ( , , ) ( )1

54 0

1

5

1

54 41

Likk

k

.20261336470055239403... 3 1

81

2

3

1

cos( )

( )

!

k

k

k

k

.20264236728467554289... 2 2

22 21

( )

( )

k

kk

= x x dx x x dxcos cos 0

12

0

1

.20273255405408219099... log log3 2

2

1

5

arctanh AS 4.5.64, J941, K148

= 1

5 2 1

1

2

1

3

1

22 10 1

1

11

kk

kk

k

kkk k

( )

( )

k

=

1 1

)(

1

1

32

1

3 n kk

kn

kkk HH

= log

( )

x dx

x2 1 21

= ( ) sin sinx x x

xdx

5

0

1

GR 1.513.7

.2030591755625688533... 1

7 4 13 4

2 1k kk

k k

( ( ) )

... ( )

log( )6 3 1

11

3 6

2

k

kk k

k k

3 .203171468376931... root of ( )x 1

... tan(log )

i ii i

i i

.2032809514312953715... log3

4

.2034004007029468657... 1 11

1

1

1

1

21

1

2

Eik k k k

k

k

k

k

( )( )

!( )

( )

( )! !

.2034098247917199633... ( ) ( ) ( )2 3 3 2 4

.2034498410585544626... 3 3

12

1

2 3 20

( )

( )(

k

)k k k

.20351594808527823401... log ( ) 31

kk

.2037037037037037037 = 11

54

1

3 91

k kk ( )

... ( ) ( )( )

2 2 326

27 2 32

k

kk

.20387066303430163678... 1024 12832

3

2

45768 2 16 3 5

2 4

log ( ) ( )

=1

41

5

46 51

1

1k k

k

k

kk

k

( )( )

61 .20393695705629065255... 5

5

.2040552253015036112... 1

51

2

53 21

1

1k k

k

k

kk

k

( )( )

.2040955659954144905... ( ) ( ) coth4 4 2 4

28

=1

41

2 4

46 41

1

1k k

k

k

kk

k

( )( )

.20409636872394546218...

3

1

12

1

6

3 3

2

3 3

2

i i

=1

6 3 13 3

2 1k kk

k k

( ( ) )

... ( )

log( )5 2 1

11

2 5

2

k

kk k

k k

.20420112391462942986... ( )4 1 1 1

21

22

4

k

kk Li

kk k

.20438826329796590155... H k

kk

( )3

1 6

... loglog

2

2

[Ramanujan] Berndt ch. 22

... e

x e dxx4

2 1

0

2

.20487993766538552923... 2 1 2 2 2 2 2 32 1

4 1

1

1

log log( )

( )

k

k k k

k

kk

.205324195733310319... ( , , )1

53 0

1

5

1

53 31

Likk

k

.205616758356028305...

2 1

2 21148

2

8

1

42

( ) ( ) cosk

kk k

k

k

= )()(2

122 iLiiLi

=log x dx

x x31

=

x x d

x

log2

0

1

1

x

= log 11

41

x

dx

x

.20569925553652392085... ( )

1

2 1

1

1

k

kk

.2057408843136237278... 11

61

kk

.2057996486783263402... 7

180

3

31

coth k

kk

...

0

33/5

sin3

4

2

13 3

dxxe x

...

2

3

1

3 32

2

20

1

Lix

xdx

log

( )

.2061001222524821839... 2

2136

49

720

1

6

k kk ( )

.2061649072311420998... ( )kk

2

2

1 1

.2062609130638129393... 15 2 13 2

2 12k k

k ik k

( ( ) ) Re ( ) Li

.20627022192421495449... ( )3 2

6

1

615 2

1

k

k kkk k

.2063431723054151902... 2 2 2 4 2 3 2 3 41

32 2

2 log log log log Li

=( )

( )

1

2 2 20

k

kk k

.20640000777321563927... ( )( )

13

5

1

5 11

13

1

kk

k k

k

k

... ( )

log( )4 1 1

11

4

2

k

kk k

k k

.2064267754028415969... 3

6 3

3

4

1

3 122

coth

kk

= ( )( )

12 1

31

1

kk

k

k

.2064919501357334736... H kk k

k

kkk k

( ( ) ) log

3 11

113 2

2

... 1

2

3

21

1 3

22

log ( )( )

k

k

k

k

Srivastava

J. Math. Anal. & Applics. 134 (1988) 129-140

.2066325441561950286... ( )3 1 1

21

2 32

k

kLi

kk k

.2070019212239866979... I

e

k

k kk

k

k

04

0

41

2 2( )( )

!

1 .20702166335531798225... 1 11

2 11

e erfik kk

( )!( )

=e

xdx

x 2

12

0

1 GR 3.466.3

5 .2070620835894383024... e

e

e

k

e k

k

1

10 ( )!

4 .2071991610585799989... 7 3

2

1

1 1

2

0

2

1

0 ( )

sinh

log ( )

( ) sinh(log( ))

x dx

x

x

x xdx

1 .207248417124939789...

1

4

1

2 10

eerfi e e erfi

e

k

k kk

( )sinh

!( )

.2073855510286739853... ( )5

5

.20747371684548153956... ( )( )

( )

12 1

2 11

1

k

k

k

k k

.20749259869231265718... 10

8

754

0

1

x xarcsin dx GR 4.523.1

... 1

22 2 1

12

log ( )( )

( )

k

k

k

k k

2 .2077177285358922614...

2

23 ))()((k

kk

.2078795763507619085... iii eei log2/ )valueprincipal(

=

2sinh

2cosh)sin(log)cos(log

iii

.20788622497735456602...

1

2

3 .207897324931068989... e e e

e

k

k

e e

k

2 1 2

0

1

2 1

/ cosh

( )!

1 .208150778890... 1

42 ( )kk

.2082601377261734183... 3

32 2 22 3

dx

x( )

.2082642730272670724... 256)3(42log192)4(3

832

2

=1

41

4

45 41

1

1k k

k

k

kk

k

( )( )

=5

24

15

1

log( )x

xdx

.2084273952291510642... e Ei Eie

x

edxx( ( ) ( ))

log log( )

2 1

2 1

0

1

.20853542049582140151... 2 2

3482

2

48

1

8

1

2

1

82log

log

Li Li 3 LHS WRONG

=log2

31 2

x

x xdx

.2087121525220799967... 5 21

2

1

7 1

2

1

kk k

k

k( )

= 11

3 1

21

0

( )

( )

k

kk k

k

k

1 .2087325662825996745... e ek e

ek

k

1 1

0

1

1

/

( )!

.20873967247130024435...

( ) Re( )

( )3 4 2 1

22 1

2

3

21

i i k

i kk

=1

41

2 3

45 31

1

1k k

k

k

kk

k

( )( )

6 .208758035711110196... e dxxsin

0

.2087613945440038371... 2 1

2112

2 2 21

1

log( )

( )

k

k k k J365

= ( )

Re( )

( )

1 3

23 22 1

k

kk

kk k

k

i

= GR 4.221.2 log log( )x x10

1

dx

1 .2087613945440038371... 2

0

1

122 2 1

1 1

log

( ) log( )x x

xdx

2 .2087613945440038371... 2

2

1122 2

1

log( )

H

k kk

k

=

log log1

1

0

1

xx dx

= log( )log1

21

x x

xdx

1 .20888446534786304666... ( ( ) ) k k

k

1 2

2

... ii ee 2log2log2

11cos)1(

=

2

1)(cos

k

kk

k

.20901547528998000945... 6

5

1

6

1

5

1

5

1

6 1 62 21

2

1

Lik

Hk

k

k

kk

( )

( )

.2091083022560555467... ( )( )

14 1

4

1

415

1

kk

k k

k

k k

=

1

2

1

2

3

2

3

2

i i

.2091146336814109663... 2

8 2

1

3

1

4 421

tanh 3

k kk

.2091867376129094839... ( )

( )!sinh

2 1

2 1

1 1

1 1

k

k kk k

k

.2091995761561452337... 2

3 31

1

2

2

0

2

( sin )

sin

/ x

xdx

1 .2091995761561452337... 2

3 3 2 1 2 2 10

2

0

k

k

k

kkk k

!

( )!!

( !)

( )! CFG D11, J261, J265

= 1

22 1

0 k

kk

k

( )

= 9

3 1 3 1

2

1

k

k kk ( )( )

=dx

x

dx

e e

dx

xx x30 0 01 1 2

sin

=x dx

x30 1

GR 2.145.3

= dx

x x

dx

x x20

211 1

... = 2 2

20

2

4 2 2 1

log log

x dx

x

.20943951023931954923...

6

2)cos(sin

15 x

xxx Prud. 2.5.29.24

.20947938452318754957...

2 12 2

1

2

)(

2

12log

k jkj

kk j

kk

k

.2095238095238095238 = 22

105

1

2 1 2 9

1

4 2 4

2

0 0

( )( ) ( )( )k k k k

k

kkk

k

.209657040241010873498... log( )csc ( )

1 2

1

k

kkk

1

.2099125364431486880... 72

343

1

82 0

1

8 82

2

1

( , , ) Lik

kk

.2099703838980124359... ( )log

!

11

k

k

k

k

2 .2100595293751996419... 10

81 3 1 1

3 2

20

1 2

3

2

300

log log logx dx

x x

x dx

x

x x d

x 1x GR 4.261.2

= log2

2 10

x

x xdx

1 .2100728623371011609...

1

02

2

12

log

2

1,2,

2

1

8

1

8

2

x

dxx

...

1

31

)()3(log

)()3(

kk k

kk

k

k Titchmarsh 1.6.2

... 4

13

6

1

4121

( )!

( )!

k

k kk

Dingle p. 70

.2103026500837349292... 1

51 ( )k kk

.21036774620197412666... sin ( )

( )!

( )

( )!( )

1

4

1

4 2 1

1

2 1 2 10

2

1

k

k

k

kk

k

k k

2 .2104061805415171122... ( ) ( )3 7

.210526315789473684 =4

19

.2105684264267640696... 6 3 220

9

1

2 10

(log log )( )

( )(

4)

k

kk k k

8 .2105966657212352544...

2

3( )

.21065725122580698811...

12

2

6

1

6

1

2 1 2 40

log ( )

( )( )

k

k k k

= x x2

0

1

arctan dx

.21070685294285255947... 1

3

1

3

13 7

1

e x

dx

xcosh

... log( )10

e kk

k

... 11

21

( )k

k k

46 .21079108380376900112... 17e

.2109329927620049189... 1

82 2 4 ( ) ( )i i 1

=

2

3

8

1

42 2

142

( ) ( )i ik

kk

= )()(4

1

8

1

2ii

= ( ( ) ) 4 1 11

kk

... 2 3 11

2

20

e Eix

e xdxx( )

( )

.21100377542970477... ( , , ) ( )( )1

52 0

1

5

1

5

1

42 21

1

1

Lik

k

kk

kkk

H

k

= ( ) log log log2 4 5 54

52

2

Li

2 .211047296015498988... 2 2 1

1

1log log ( ) /

k k

k

1 .2110560275684595248... E(¾)

.21107368019578121657... ( )

!/3 1

11

1

2

3k

ke

k

k

k

7 .2111025509279785862... 52 2 13

...

11

2 1

21

( )

( )!!

k

k kk

!! J166

.2113553412158423477...

( )3

21

kk

k

.2113921675492335697... 1

21( ) Berndt 7.14.2

.2114662504455634405... ( )log

3 1 11

13

21

3

2

k

k kmm

kk

mk m

=

log cosh log logcosh

1

3

3

23

3

2

=

2

35log

2

35log

ii

.2116554125653740016...

e k

k

e

k2 1

sin

.2116629762657094129... 14

coth

8 .21168165538361560419... Ei e( )

... 2 4 32

3

23

3

162 2

4

7 3

4

2 3 2

GGlog

( )

= H

kk

k ( )4 1 21

... ( )

!

!

1

23

k

k k

.21184411114622914200... 8 216

3

1

2 2 80

log( )

kk k

... 1

42 2

0

1

log ( ) log sin cosci x x x dx

.2118668325155665206... 2 1 11

11

e e ee k kk

k

( ) log( )( )

J149

.21192220832860146172... 3 1

318

41

1 2

Gk

k

k

k

( )

( / )

1 .21218980184258542413... 1

3 1 30k

k

/

.21220659078919378103... 2

3

0

3 23

0

1

/

sinx x dx

.21231792754821907256... log log31

22 2 3 2

1

2

dx

x x

... 6 32

3

0

( )

x dx

e ex x

=

0 13/1xe

dx

.2123457762393784225... e

.21236601338915846716... ( )2 1 1 1 1

21

22

k

k kLi

kkk

k

.21243489082496738756...

( )k kk

k 2 11

.2125841657938186422...

2

2

1 42 20

2e

x dx

x

x dx

x

cos cos

AS 4.3.146, Seaborn Ex. 8.3

= sin( GR 3.716.6 tan ) tan/

20

2

x x d

x

= cos( tan ) sin20

x xdx

x

GR 3.881.4

.2125900290236663752... log k

kk3

1 1

.2126941666417438848... log log2 21

2 42

31

H

k kk

k

.21271556360953137436... 10 3 24 2

108

1

3

2 1

3 21

log ( )k

k k k

.2127399592398526553... Ik kk

3 0 10

21

64 1

1

3( ) (; ; )

!( )!

F LY 6.114

.21299012788134175955... 2 2 6 4 21 2 1

2 2

1

1

log log( ) ( )!!

( )!

k

kk

k

k k

=

1

1 2

8

)1(

kk

k

k

k

k

.2130254214972640800... ( ( ) ( )) k kk

1 2

2

.2130613194252668472... 2

11

2 1

1

1e k

k

kk

( )

( )!

.2131391994087528955...

02 )22)(12(4

)2(

4

)3(7

kk kk

k

Yue & Williams, Rocky Mtn. J. Math. 24, 4(1993)

...

8

1

2

3

29 42 2

0

2

ee erfi erfi e x x dxx

/ sin cos

.2132354226771896621... hgk k

k

k

kk

k

( ) ( )( )1

7

1

71

1

721

1

1

.21324361862292308081... 1

7

1 .21339030483058352767... 1

22

1

2 111

1k

kk

k

½

3 .21339030483058352767...

111

0

2

)(

2

)(

kk

kk

kDk,

1

0 )()(k

knD

.21339538425779600797... 4

9

2

31 2

0

1

log

logx x x dx

x

GR 4.241.10

GR 4.384.11 log(sin ) sin cos/

x x x d2

0

2

.2134900423278414108...

sin ( )

!( )

4

2

1 41

120

k k

k k k

.21355314682832332797... 2

3

1879

7560

1

3 3

1

1

log ( )

( /

k

k k k )

.21359992335049570832... 1

82 2 2 2

1 2

2 1

1 2

1

( cos sin )( )

( )!

k k

k

k

k

=

1

211

)!2(

2)1(

k

kk

k

k

.2136245483189690209... 1

6

3

2 3

1

32 21

cot

kk

.2136285768839217172...

111 )92(!

1,2

13,

2

11

11

1

k kk

kF

511

32

945

641

eerfi

.2137995823051770829... 1

9

8

27

1

2 2

1

22

1

1

arcsinh( )k

kk k

k

.21395716453136275079... 1

6

3 1

641 1k k

k

kk

k

( )

.2140332924621754206... 5 5 15

515

2

k

k k

kk

( ( ) )

1 .2140948960494351644...

233/23/1

11

))1(())1((2

1

2

3cosh

2

1

k k

.2140972656978841028... 1

1

1 1

1 1e e e

dx

ekk

x

1 .214143334168883841559...

( ) ( ) ( )k k kk

12

.2142857142857142857 =3

14

1 .2143665571615205518... ( )

log

k

kk

1

2

.2146010386577196351...

1 2

2

1

8

3

2

1

2

3

2

1

2

1

2

1

2

log

=( )

1

232

k

k k k

.21460183660255169038... 14

1

2 30

( )k

k k

=( )!!

( )! ( )

2 1

2 4 121

k

k kk

J387

=dx

x x

x dx

x

x

xdx4 2

1

2

20

4

20

1

1

sin

cos

arcsin

( )

/

2 .21473404859284429825... 3

14

1 .2147753992090018159... log log log ( )2 3 3 33

4

x dx GR 6.441.1

.2148028473931230436... ( )

log2

81

1

812

1

k

k kkk k

.21485158948632195387... 8 4 8 5 21

4 2 40

log log( )

( )

k

kk k

.21506864959361451798... ( ( ) )

!

k

kk

1 2

2

.215232513719719453473... ( )

1

2 1

1k

pp prime

, where p is the kth prime

.2153925084666954317... ( )( ( ) )k kk

2

2 1

... dxx

xdxxx

4/

02

34/

0

2

tan

sincossin

26

1

3

1

.2156418779165612598... 24 62

22 2

0

1

arcsin arccosx x dx

... 2

32

7 3

4

4

27

4

2 1

( )

( )

k

kk

= ( ) ( )( )

1 11

22

kk

k

k kk

.2157399188112040541... 7 331 5

4 6 1 2

4 2

51

( )

( )

( /

)

k

kk

.2157615543388356956... 3

4

4

31

31

1

log ( )

k kk

k

H

1 .2158542037080532573... 12 2

22

( )

2 .21587501645954320319...

( )

( )

k

kk

1

1 12

2

.21590570178361070... 1

2

1

2 2

1

0

e e

k e

k

k

( )

... ( ) ( )k kk

1 11

.216166179190846827...

1

12

)!2(

2)1(

4

1sinh

4

1

k

kk

ke

e

1 .2163163809651677137... cos sin sin( )

( )

1

4

1

41

1

21

1

2 10

k

k

=( )

( )!!

( )

( )!!

1

4 4

1

4 2 2 40 0

k

kk

k

kkk k

13 .2163690571131573733... k kk

4 2

2

1( ( ) )

.2163953243244931459... 3 3 3 2 11

3 11

log log( )

kk k

GR 1.513.5

= ( )

( )

1

2 11

k

kk k k

= log 120

1

x

2 .21643971276034212127... k

k kk k

2

21

21

3

21

1

2

.2167432468349777594... 2

323 2 2

1

8 7 8 31

log( )( )(k kk )

J266

1 .2167459561582441825... log( )(

26

1

4 3

1

1 4 121 0

)

k k k kk k

... 1

2

2

2

1

8

1

2

1

2

log i i i i

= cos

)

2

0 1

x

e edxx x

.216872352716516197139... 5

4log

5

4

2

5log5log2log25log2log2)3( 2

322

Li

5

4

5

133 LiLi

= H

kk

kk 5 2

1

.2169542943774763694...

sin

2

21

22

2 k

... = 93 5

2

4

0

( )

sinh

x dx

x

.21740110027233965471... 2

21

9

4 1 3

k

k kk ( )( )

1 .2174856795003257177... 24 2 5 1 241

44 0 10

Ik kk

( ) (; ; )!( )!

F

.2178566498449504... ( )

1

432

k

k k

.218086336152111299701... 1

2 242

1

2

2 1

1

log

( )

( )

kk

k

H

k k

2 .2181595437576882231... 2

5

1 .21816975871035137... j g2 22 J311

.2182818284590452354... ek k k kk k

5

2

1

2

1

2 41 0( )! !( )( )

1 .218282905017277621... arctan( )

( )e

e

k

k k

k

1

2 1

2 1

0

.2185147401709677798... ( ) log3 48 22

38 64

1

4

2

4 31

k kk

= ( )( )

13

41

1

kk

k

k

37 .2185600000000000000 =116308

3125 4

2

1

F kkk

k

3

1 .2186792518257414537... 2 2

2 2

sinh

2 .21869835298393806209...

2

1

1arcsin

kk

...

12

11log

1arctan

21

)12(

)()1(

kk

k

kkkkkk

k

.2187858681527455514... Hk

kk 6

6

5

6

51

log

.2188758248682007492... 2 5log

... 136

625

4 2

625 51

2

4

1

arccsch( )k

k

kk

k

... cosh( ) ( ) sinh

1

2 81 1

12

14

erf erfix

dx

x

.219235991877121... ( )

1 1

31

k

k k k

1 .21925158248756821... 8 2 4 2 2 32 1

22

1

log log ( )( )

H

k kk

k

.2193839343955202737...

Eik k

k

k

( )( )

!1

1

1

LY 6.423

= logdx

e

e

xdxx

x

1 0

1 1

= dx

ee x

0

2 .2193999651440434251... ( ) ( )3 6

.2194513564886152673...

12 2

1

2

11

kkLi

k

9 .2195444572928873100... 85

.21955691692893092525... 2 22

1

4 4 21

( ) coth

k kk

= ( )( )

12 2

41

1

kk

k

k

=

Re( )

( )

k

i kk

2

21

.21972115565045840685... e

e x

dx

x x

dx

x4

5

4

1 1

2

12 7

14

1

cosh cosh

.2197492662513467923... ( )( )

!

1 1 0

1

k

k

k

k

.2198859521312955567... 1 1 10

1

ci x x x dx( ) cos log cos

.219897978495778103... ( ( ) ) kk

2

1

2 1

2265 .21992083594050635655... 3

4

7

.2199376456133308016... ( ) ( )

1

4

1 3

1

kk

kk

H

k

... 3 2 2 2 2 3

6 24

2

2

2

24

2

2

2

6

3

3

log log log log ( )

= log sin2 2

0

x x

xdx

.22002529287246675798... 1

21

1

41

1

21

Jk

k

k

kk

( )( )

( !)

... log21

41

2 31

2 3

i i

1

4

3 3

6

3 3

6

i i

= ( )

1

3

1

31

k

k k k

.2200507449966154983... 2 24 2 1

2

1

2

2

1

log( )!!

( )!!

G k

k kk J385

... ( )

log2 1

21

1

2

1

222

1

k

kk

k kkk k

.2203297599887572963... 2 4 2

1

2 4 2

1

2coth coth tanh

=1

4 4 2

2

221 1k k

k

ikk

k

Re( )

( )

= ( )( )

14

4

1

4 114

1

kk

k k

k

k

1 .2204070660909404377... ( ) ( )3 4

.2204359059283144034... 27

4

3

1

6

1 2)1(

= 3/22

3/12

3/22

)1(3

1)1(

2

)1(

108

5

LiLi

3/22

3/2

)1(4

)1(

Li

= 1

36

1

3

5

6

1

3 2 20

( )

( )

k

k k

=

02

1

03

13 1

log

1

logxx ee

dxxdx

x

xx

x

dxx

...

02 12

3

28

33/2xe

dx

.22056218167484024814... 1

22

1

22 1

1

2

1

1eEi

kH

k

kk

kk

log( )

!

.2205840407496980887... cos Re

2

2 i

.220608143827399102241... 3 3

24 2

36 1

22

0

1 ( )log log( ) log x x dx

...

( ) log( )

log log2 4 23 3

21

1

0

1

2

x

x dx

... log

tan tan2 1

20

1

0

1

x xdx x x dx

2 .220671308254797711926... arctanarctan log

log2

12 24 4 2 3 3

2 32 2 2

2

20

x

x xdx

1 .2210113690345504761... 11

22

( )kk

.2211818066979397521... 1

1627 3 3 2 27 3 8 32( log ( ))

=1

3 231 k kk ( )

.2213229557371153254...

21

6

205

1445 1 2 5 2 5 ( ) ( ) ( , , ) ( , )

1 .2214027581601698339... e5

2 .22144146907918312350... 2 14

dx

x Marsden p. 231

= x dx

x

2

4 1

=d

1 20 sin

Marsden p. 259

= log cosh

cosh

x

xdx

xdx

x2 12

2

2

00

= log 11

2 40

xdx

.22148156265041945009...

122

2

1)(11

1

kkk kk

k

.22155673136318950341...

8

... 6 51 3

0

1

( )log( ) log

x x

xdx

... ( ) ( ) ( )( ) ( )3 11

42 22 2 i i

=x x

e edxx x

2 2

0 1

sin

( )

.22157... log ( )!

( )!

2

2

1

8

2 4

2

2

2

k k

kk

= ( )!!

( )! ( )

2 1

2 122

k

k kk

.221689395109267039... 1

13 1 3 1

13

2 16 3

2kk

k kk k k

( ( ) ) ( )

=1

31

1 3

2

5 3

2

1 3

2

5 3

2

i i i i

=

1

3

1

61 3

3 3

21 3

3 3

2( ) ( )i

ii

i

=

3

2

3 3 3

5 3

2

1

3

2

3 3 3

5 3

2

i

i

i

i

= 1

31 1 2 1 1 2 12 3 1 3 2 3 2 3 ( ) ( ) ( ) ( )/ / / /

.2217458866641557648... 1

63 4 2

2

41

(log log )( )

kk

k

.22181330818986560703... 2 2 26 2 1

22

20

log log( )

k

kkk

1 .2218797259653540443... log

kk

kk

112

2

.221976971159108564... 1

1142 kk

.22200030493349457641... ( )

!

/2 1 1 1

1

1

2

2

k

k

e

k kk

k

k

.22222222222222222222 =2

91

21 2

0

1

1

( ) arccoskk

k

kx xdx

=

1 19

3)(

kk

kk

...

1 )!1(k

k

k

B

... 1

6

2

3

2

3

1

21

21 2 1

1

log arctan ( )

k kk

k

H

2 .2226510919300050873... 3

4

1

2

1

2

2

0

e

e

k

kk

( )!

.2227305519627176711... 21 3

4 16

12

2 1

4

41

( )

( )

k

kk

= ( )( )( ) (

)

11 2

21

3

kk

k

k k k k

... G log2

2 .22289441688521111339...

8 4 1

1

2 21

2

4

=8

8

2 1

231

11k k

k

kk

k

( )

8 .22290568860880978... 2 1

1

2

1

k 1

k

k

k

k

k

e

k

( )

!

/

2 .2230562526551668354... 1

29 3 3

3

3 321

22

( log )( )

k k

k

kk

k

=1

1 230 ( )(k k )k

.2231435513142097558...

1

1

11 4

)1(

5

1

5

1Li0,1,

5

14log5log

kk

k

kk kk

= 21

2 1 92

1

9 42 10 0( )k

dx

ekk

x

arctanh1 K148

.2231443845875105813... H

kk

kk

( )

( )

3

1 3 1

.2232442754839327307... 2 1 1 21

2 1

1

2 1 2 2

1

1

1

1

sin cos( )

( )!( )

( )

( )!( )

k

k

k

k

k

k k k k

= x xd2

0

1

sin x

=log sin log

sin2

21

41

1x x

xdx

x

dx

x

e

.22334260293720331337... ( ) ( )

( )!

2 2

21

k k

k

1

k

.2234226145863756302... ( )

!

1

10

k

k k

.22360872607835041217... ( ( ) ) k

kk

1 2

2

98 .22381079275099866005... 267 1 9

0e

k

k

k

k

( )

!

.223867833455760676... 1

4 2

1 1

4 22

2 2

e e

csch

=( )

1

2

1

21

k

k k J125

.2238907791412356681... Jk

k

k

k

k

k

k0 0 1 2

0

1 2

21

2 1 11 1

( ) (; ; )( )

( !)

( )

( !)

F LY 6.115

= ( )( )!

( )( )!

11

2

21

2

2

0

12

0

k

k

k

kk

k

k

k

k

k

k

.22393085952708464047...

2

2

12223

)1log(1)12(

)1(

4

kkk k

k

k

k

kk

.22408379...

11 1

)(

)1(

)(

kk k

k

kk

k

3 .2241045595332964672... ( )( )

! /

11 3 13

2

2

1 32

k

kk

k

k k

k k

k k

e k

.224171427529236102395...

3 82 1

2 11

log( )

( )

k

k kk

... ( )( )

12 1 1 1

22

2 21

k

k k

k

k kkLi

k

.2244181877441891147... 7

361

7

6 71

( log )

k Hkk

k

.2244806244272477796... )1(22

2log

33

5

= ( )

logk

k kk k

kk k

1 1

3

1

3

1

21

1

1

2

2

... e

.2247448713915890491... 3

21

2 1

2 31

( )

( )!

k

k kk

!!

1 .2247448713915890491... 3

2

1

12

2

0

kk

k

k

= 11

6 30

( )k

k k

... 8 1

51e

ex

dx

x

sinh

1 .2247643131279778005... 1

18

3

6

1

92 21

cot

kk

.2247951016770520918... 1

2 4 2 2 2

1

8 1

2

821 1

cot

( )

k

k

kk

k

.22482218262595674557...

2 3 3

1

3 1

2 1

322 1

cot( )

k

k

kk

k

.224829516649410894... 1

8132 3 24 2 27 3

1

2 32

31

( log ( ))( )

k kk

.225000000000000000 = 40

9

.22507207156031203903... 1

0

3arctan2

2log

324dxx

.225079079039276517... 1

2

.225141424845734406... 22

2 11

2 20

1

G Edk

k

log ( )k GR 6.149.1

.2251915709941994337...

1

1

!)!1(

2)1(

k

k

k

.2253856693424239285... )()(14

1

16

)3(3333 iLiiLiLi

=

1

02

2

13

2

1

loglog

x

dxxx

xx

dxx

=x dx

e x

2

20 1

1 .225399673560564079... 2

2 1

1

20

e

e e kk

( )

1 .22541670246517764513... 3

4

1 .22548919991903600533...

2

2

1

1

coth(log )

... 1

184 2 2 2 3 2 2 1

21 2

1

arccot log log ( )k kk

k

kH

.2256037836084357503... H k

kk

2

1 8

1

74

8

72

4 2

4 2

log log

1 .2257047051284974095... ( )31

kk

.2257913526447274324... log ( ) log( )

21

2

222 1

1

k

kk

k

kk

k K ex. 207

.226032415032057488141... 8

1

6 51

arctan x

xdx

3 .22606699662600489292... ( ) ( ( ) )

1 21 3

1

k 1k

k

...

42

4

2 10

erfik k

k

k

!( )

.2265234857049204362... e

12

2 .2265625000000000000 = 285

128

1

54 0

1

5 54

4

1

, , Lik

kk

... 1

4

5

44

0

4

x e dxx

.226724920529277231...

8 3

.22679965131342898... 287

4

153

22

77

22

2 1 22

3

1

3

log log( )( )(

k

k k kk

k )k

H

.226801406646162522... 2

7

84 2 2

3

8 8cot log logsin logsin

=1

8 21 k kk

.2268159262423682842... 1

972 2 168 2 149

2 42

1

log log( )

H kk

k k

1 .2269368084163... root of ( )x 5

... 2

272 3

2

3 2

2

31

, ,log x

xdx

.22740742820168557... Ai(-2)

.22741127776021876... 3 4 21

2 2 3

1

10

1

2 21

log( )( )

( )

( )kk

k

kk k k k

= ( )!!

( )! ( )

2 1

2 1 21

k

k kk

1 .22741127776021876... 4 4 21

4 1

22

0

log( )k

k k

k

k

=1

121

22k k

k

k

kk

k( ½)( )

( )

=( )

( )

k

k k

!

!k

1

2

121

... 23

2

1 32

1

21

arcsinh

( )k k

kk

k

k

4 .2274535333762165408...

2

3 21 4

1 4

1

4log

( / )

( / )

1 .2274801249330395378... 3 5 66

372

12 22 1

1 1 12 42

12 3 4

( ) ( ) log

H

k k k kk

k

1 .22774325454718653743... H

k kk

k ! 21

1 .2278740270406456302... 1

11 k kk ! ( )

1 .227947177299515679... log( ) ( , , )2 21

21 0

... ( )( )

arctan

12 1

1 3 1

32 1

k

k k

k

kk

k

k

k

2 .2281692032865347008... 7

16 .2284953398496993218... e erfk

k

k

2

0

12

22

!!

= 8

35

1

1

2 2

2 4

p

p pp prime

...

2

)(

2 2 2

)(2

)1(

221

)2(

)(

k

k

k s ks

k

kkkk

k

.22881039760335375977... )1,2,2(1

)2,3(2

)5(11)3(

2 123

2

MHSjk

MHSjk

... dxxx

x

1

34

222 )1log(

48

7

24

2log3

.2289913985443047538... G

k

k

k4

1

4 2 20

( )

( )

=1

8 6

1

8 22 21 ( ) ( )k kk

=x dx

e e

x dx

x xx x2 20

3 11

log

=x x

xdx

log

1 41

.2290126440163087258... e

ee

H

e

kk

kk

11 1

1 1

1

(log( ) )( )

.22931201355922031619... H

kk

kk

(2)

51

... log

( ) ( )

2

21 1 1

x

x xdx

.229337572693540946...

1

21

1 1

21

loge ek

k k

.2293956465190967515... I e ee

k

k

k0 0 1 2

0

2 1( ) (; ; )( !)

F

... 3

22 1

11

2

log ( ) ( )k

k

k

kk

= log 11 1

12

k kk

... 1

8

1

2

1

4

1

2

1

2 1 4

1

1

cos sin( )

( )!

k

kk

k

k

.229637154538521829976... 3

22 1

11

2

log ( ) ( )k

k

k

kk

.22968134246639210945... 1

2 2

1

2

1

2 2

1

1

sin( )

( )!

k

kk

k

k

... sincos ( )

( )!2

1

1

1

2

1 1

2

1

2 2

k

k k GR 1.412.1

= sin1

21

3x

dx

x

2 .2299046338314288180... ( )

( )!cosh

2

2 2

1 1

12

1

k

k k kk k

.2299524263050534905... 24

25

3 2

5 10

1

4 51

log

( )

k kk

.2300377961276525287... 4

2 1

2 1 2 20

1

x x

x

arccos

( )

dx GR 4.512.8

.2300810595330690538...

1

4

5

21

5

2 1 21

cos( )

kk

.23017214130974243... 1

2 2

1

2 2

12

21 8

1

1eerfi

kk

k

k

kk/

( )

!

... 1626 1

2 30

e k k

k

kk

( )

! ( )

.2302585092994045684... log10

10

3 .23027781623234310862... 1

22

2

1

4 120

csch

( )k

k k

.23027799638651103535... 1

4 21k

k k( )

...

1

2 3102k j s

jks

.23032766854168419192... 5 5

121

1

2

21

1

( )k

k k

k

k

.23032943298089031951... 1

6

... 1

4

1

2

1

2 2

i i i

2

i

= cos

)

x

e edx

x x

10

...

log

cos2

2

1

1 2 20 x

xdx

x

.230769230769230769 = 3

13

5

0

sin x dx

ex

9 .2309553591249981798... k

kkk

4

1

.2310490601866484365... log ( ) ( )2

3

1

3 3

12

20

1

1

k

k

k

kkk k

kk

J292

= dx

x x

dx

x x

dx

e x41 0

203 1 3 2

( )( ) 3

1 .23106491129307991381... cosh cosh sinh sinh cosh sinh 2 2 2 2

= eii

.231085022619546563... ( )

!

k

k kk2

2

... log ( ) 2 11

kk

.2312995750804091368... k

kk

!!

( )

! 3

0

1 .23145031894093929237...

2

1122 2 1 2

1

2

log ( log )( )k H

kk

kk

.23153140126682770631... 1 1

1

1

1 e

k

k

k

kk

/

( )

!

.2316936064808334898...

2

1Ai

... k

kk

kLi

kk k

11 1

1 12

22

2

( ) log

2 .23180901896315164115... 1

21

2

( )kk

.2318630313168248976... 2 2 21

20

log( )

kk

k

1 .2318630313168248976... 2 2 2 121

log( )

k kk

k

2 .2318630313168248976... 2 2 2 21

20

log( ) k k

kk

1 .23192438217929238150... ( )( ) ( )

12 1

1 1

1

kk

k

k k

.2319647590645935870... 9

212

2 80

e k

k kk

!( )

.23225904734361889414... 6

2

91 2 2

0

1

K x x( ) arcsin( )dx

.2322682280657035591... 351 14 3 126 3

588

1

3 71

log

( )k kk

.2323905146006299... ( )log

( )

112

k

k

k

k k

1 .2329104505535085664...

1

0

2

1

log)3,3(16)1)3((16

x

dxx

3 .2329104505535085664... 16 3 11

2

0

1

( ( ) )log

x dx

x GR 4.512.8

= 16 3 181

2

2 20

( ) / /

x dx

e ex x

...

1

2)12(

51

2

5cos

4

1

k k

... 2 2 1

2

3

2

1

21 2

i ( / )

.2331107569002249049... ( )3

32

3

... 4 48

2

2

2

0

1

log

arctan logx x dx GR 4.593.1

.23333714989369979863...

( )2

41

kk

k

1 .2333471496549337876... Im ( )( ) 2 i

1 .2334031175112170571... arcsinh2

.2337005501361698274... 2

22 28

11

2 1

1 1

2

( )

( )( ( )

k

k k )

kk

k

=( )!( )!

( )!

k k

k

k

k

1 1 2

21

= dxx

xx

1

02

2

1

log

=log x

x xdx4 2

1

1 .2337005501361698274... 2 2 2

2 32 08

1

1 2 1

1

k k

k

k

k kk k

( )

( )

!

( )!!( ) J276

= ( )( )

( )( ( ) )2

3 2

4

1

2 12 12

0

2

1

kk k

k k

= 1

4 1

1

4 32 21 ( ) ( )k kk

= 2

221

k

k k

kk

=

1 1 2

)(

j kjk

jk

= log x dx

x20

1

1 GR 4.231.13

= log x dx

x40 1

= arctan x dx

x1 20

= arctan x dx

x

2

20 1

GR 4.538.1

= arcsin x dx

x1 20

= K kdk

k( )

10

1

GR 6.144

= E k GR 6.148.2 dk( )0

1

.2337719376128496402... ( )

!

( )

( )!!/k

k

k

ke

kkk k

k

k2 21

1

22 1

1 2

1

... sin1

31

2x

dx

x

.23405882321255763058... 1

2 2 2 2

1

21

1

21

1

2

cot

=k

k k

k k

kk

k

1

2

2 2

231 1

1 ( ) ( )

.2341491301348092065... 1

6 1 61

0

1k

kk

k

k

( )

.23416739426215481494... ( )3

6

1

36 616 3

1

k

k kkk k

.23437505960464477539... ( )

!

1

20

k

kk

.23448787651535460351... ( )2 1

2 1

11

1 2

k

kk

kk k

arctanh

.23460816051643033475... log k

k kk3

2

.23474219154782710282...

( )3

6

1

21

1

61

1

6

=1

36 6

2 1

65 31 1k k

k

kk

k

( )

.2348023134420334486... 1 11

40

1

21

Jk

k

kk

( )( )

( !)

.2348485056670728727...

4

12

4116

16 43

2

1

,( )kk

.23500181462286777683... 3 2

2

5

3 31

2log log

dx

x x

... 2 1

5 4 11

21

1

sin

cos( )

sin

kk

k

k

.235028714066989... 1

3612 3 2 3

3

2

3

22

sin csc sec

=( )

1

34 22

k

k k k

.2352941176470588 =4

17

4

0

sin x dx

ex

=1

2 41 2 1 2 2

0cosh(log )( ) ( ) log

k k

k

e J943

1 .2354882677465134772... 33

2 1

3 13

32 1

sech

k

k

k

kk k

exp( )

= exp logk

kk

3

32 1

=

13

)(

k k

kz

= 3 1 15 3

2

5 3

21 3 2 3 ( ) ( )/ /

i i

... 3 3

42 2

133

20

1 ( )log

log ( )

x

xdx

.23550021965155579081... 1

2 2 1

1

2 11

1

11

k kk

k

kk( )

( )

.2357588823428846432... 2 1

2

1

2

1

1

1

1

1

1 2

1

1e k k

k

k k k

k

k

k

k

k

k

( )

!( )

( )

( )!

( )

( )! !

.23584952830141509528... 89

40 CFG D1

.235900297686263453821...

5

15log5log2log42log4

2

1

4

12

222 LiLi

=H

k kk

kk

k

kk5

1

41

1

21

( )

.2359352947271664591... ( )( )!

/11

1

1

e e ek

k ee

kk

... 5

3 1 6 50

dx

x /

.23605508940591339349... 1 28

2 2

2 2 0

4

log log tanlog

sin log(sin )/

x x dx

= 1 2 2 12 2

2 2

log ( )

log

.236067977499789696... 5 21

3

1 .236067977499789696... 5 11

2 1

21

1

( )k

k k

k

k

=

( )

( ) !

3 1

2 20

k

k k kk

8

Berndt Ch. 3, Eq. 15.6

2 .236067977499789696... 5 2 1 21

5

23

0

kk

k

k

4 .236067977499789696... 3 2 1

... 12 2 9 3 3 36

1

6

2

3 21

log log

k kk

2 .236204051641727403...

( ) ( )2 5

22

224

277 3

1 .2363225572368495083... 1

4 2 6

2

3

1

3 221

cotkk

.23645341864792755189... 3 2 1 2 11

2 1

1

1

cos sin( )

( )!( )

k

k k k

1 .2365409530250961101... 3

4 2 2

2

0

2

0

e k

k

k

kkk k

! ( )!!

... csc ( )3

6

1

18

1

9

1

2 20

k

k k

.2368775568501150593... 1

216

5

6

1

32 2

2

20

( ) ( )

x xdx

e ex x

=log2

31 1

x

xdx

.2369260389502474311... 25

64

1

2

1 6

0e

k

k

k

kk

( )

!

... 9 3

4 1

2 3

31

3

30

1 ( ) log

( )

log

( )

x

x

x x

x 1

.2374007861516191461... log( )3 3

.237462993461563285898... 2

4

3

1

5 20

( )

/

k

k k

=

2/

02

2

)sin1(

sin

dxx

x

... 252

15log

4

3

24 22

2

Berndt Ch. 9

9 .2376043070340122321... 16 3

3 CFG D11

17 .2376296213703045782... k

kk

3

1 1( ) !!

.23765348003631195396... 2

2130

137

1500

1

5

k kk ( )

.2376943865253026097... ( ( ) ) jkkj

3 111

98 .23785717469155101614... 24 212 1

22 3

30

log( )

e x dx

e

x

x GR 3.455.2

.2378794283541037605... 1

144

1

6

2

31 1

3 31

( ) ( ) log

x dx

x x

... ( ) ( ) ( )

( )

( )2 3 2 2

1

1 21

k

k kk

.23799610019862130199... 113

2 3k ks d

k log( )

s

.2380351360576801492... ee

erfk

k

k

2

1

2

1

0

( )

!!

.238270508910354881... 8 21

2

146

15

1

2 2 70

arctanh

kk k( )

6 .238324625039507785... 9 2log

.2384058440442351119... 2

12e

16587804425989... = 2 2 2 3

20

2 3 2

16 2 2 1

( log ) log

x dx

x

1 .2386215874332069455...

1

/1

1

1 2

1!

)2()1(

k

k

k

k ek

k

.2386897959988965136... ( )

1

120

k

k k k

.23873241463784300365... 3

4

.2388700094983357625... 1

16

3

2 1

3

1

( )( )

ee

k

k !k

2 .2389074401461054137... k H k

kk

( )3

1 2

2 .23898465830296421173... ( ) ( )3 5

.2391336269283829281... 2 1 11

2 2 31

2 2

200

cos sin( )

( )!( )( )

( )

( )!

k

k

kk k k

k

k

=log coslog2

1

x x

xdx

e

.2392912109915616173... Ei ekek( ) log( )

/ 1 1 1

1

.23930966947430038807... ( )k

kk

3

32

= 1

1

1 13

26 3

212 3

2k k k kk k k

k

= ( ) ( ) ( )3 1 9 3 1 9 3 11 1 1

k k kk k k

.2394143885900714409... 1

8

2

4

1

42 21

cot

kk

2 .2394673388969121878... 2 3 2 4 ( ) ( )

... 1 2

3 6 41 3

43 3

logcot ( cot (

= ( )

12

2

k

k k k

.2396049490072432625... log( )

e

e2 1 J157

... 12 3

3

1

2

1 22

2 1

1

1

arcsinh( )

( )

k k

k k

kk

.2396880802440039427... 1

831 kk

.2397127693021015001... 1

2

1

2

1

4 2 1

1

1

sin( )

( )

!

k

kk k

...

1 )(

1

k k k

.2397469173053871842...

( )log

32

31

k

kk

.2397554029243698487... 21

2 2

1

2 22

1

1

sin( )

( )!

k

kk k

.23976404107550535142... 1

22 2

1

23 2

1 2

22 0 10

Jk k

k k

k

( ) (; ; )( )

!( )!

F

.2398117420005647259... GR 4.381.1 ci x x dx( ) log sin10

1

= ( )

( !)

1

2 21k k k

k

.239888629449880576494...

1 2

1)2()2(12log

4

7

k k

k

=

kk

kk

42

2

2

11 1

2log

... dxx

x

1

06

22

)1(

log

6

2log5

303

1

.2399635244956309553... 1

2

1

4,

.24000000000000000000 =6

25

1

61 0

61

( , , )k

kk

.24022650695910071233...

11

1

12

)1(21)1(2log

2

1

kk

k

k

kk

k

H

k

H

= dxx

x

1

0 1

)1log( GR 4.791.6

=

0

2

1

1

0 1

loglog

1

)1log(xe

xdx

x

xdx

x

x

.2402290139165550493... 2 1 2 2 11

10

1

log( ) log e

e

edx

x

x

.24025106721199280702... 3

25

.2404113806319188571... ( )3

5

.2404882110038654635... ( )( )

13

13

1

k k

kk

H

1 .2404900146990321114... 1

1 3

1

2 3/ /

.24060591252980172375... 1

2

1

2

1

4 2 1

22 1

0

arcsinh

( )

( )

k

kk k

k

k

.2408202605290378657... 1 2 2 32 3

1

( ) ( )( )

k

kk

= ( ) ( ( ) )

1 11 2

1

k

k

k k 1 Berndt 5.8.5

.2408545424716093105...

12

21

8k k

k

... 1

1 21 ( )! (k kk

) Titchmarsh 14.32.1

... 2 24

23 3

16

22

0

1

log( )

arctan logx x dx

... 1

6

2

3

1

1 2

1 12

1

( ) ( )!

( )!

k

kk

k

k

13 .2412927053104041794... 257

32 2

5

1

e k

k kk

!

... 2 2 3

2112

2

2

2

6

3

8 2

log log ( ) ( )k

kkk

.2414339756999316368... 5 2

4

5

8 2 22

log

( )

1

k

kkk

.24145300700522385466... 1

1

.241564475270490445... log( )

log

4 1

1

2

20

1

x

x

dx

x GR 4.267.2

=x x x

x xx dx

log

loglog( )

112

0

1

GR 4.314.3

.24160256000655360000...

( )

( )

2

5 1

1

512

1

kk k

k k

... 15 33

6

Associated with a sailing curve. Mel1 p.153

...

2

20

1

1

( )

( )sin/

e

ee xx dx

))

.24203742082351294482... ( (22

kk

1 .2420620948124149458... 1

2 1

2

2 11 2

21

1

11( )log log( )

( )

`k

k

k

kk

kk

kkk

k

=

1

1

2

)(

kk k

k

.242080571455143661373... logsin/3

32

27 8 8

50

x

xdx

.24214918728729944461... 112 9 24 2

271

2

0

1

loglog( ) logx x x dx

.242156981956213712584... ( )( )

14 1

1

1

kk

k

k

.2422365365648423451... 3 2

2 20

2

3 11128

x dx

e e

x dx

x xx x

log

.24230006367431704... ( )

1

230

k

k k

.242345452227360949... c231

63 2 3 ( ( ) ( ) ) Patterson Ex. A4.2

.24241455349784382316... ( )

!

1 1

1

k ek

k

H

k

3 .24260941092524821060... 1

2 (log )k kk

.24264068711928514641... 3 2 48 1

2

1

k

k

k

kkk ( )

4 .24264068711928514641... 18 3 2

...

1

11

kk k

k

k

kLi

2 .24286466001463290864... 8 13 38 3 3

12

4 3 2

3 31

1

( )( )

k k k k

k kk

= ( ) ( ( ) ( ) )

1 23

1

k

k

k k k

.24288389527403994851... 2 3 3( ) ( )

...

1

21

21

)!12(

)!()!(

23 k k

kk

= 1

0

32

2

)arctan(1

dxxx

x

.24314542372036488714... 1126

9 7 5

2 2

9

1

2 921

log

( )k kk

2 .2432472337755517756... 128

1

48

7

120

24

= k kk k k

kk k

3

1

2 4 2

2 42

2 11

1( ( ) )

( )

( )

.2432798195308605298... e

e

k

kk

k

2

1

12

0( )

( )

!

B

.24346227069171501162... ( ) ( )

!/

/k k

ke

e

k kk

kk

k

1 11

2

11

21

.24360635350064073424... 5 82 21

ek

k kkk

! ( )

.2436903475949711355... 9G

.2437208648653150558... 83

23

1

21 3

1

2 30

log ( , , )( )

( )

k

kk k

.2437477471996805242...

4 2

1 2

2 2

1

4 3 104

2

40

1

1

log( ) ( )k

k k

dx

x

x dx

x 1

=dx

e ex x30

.243831369344327582636... 7

2

1

2

3

2

3

2

3

22 1

1

cot ( )k

k

k

.24393229000971240854...

k kk kk k

( ( ) ) 3 11

113 3 2

2

.2440377410938141863... ( )1

.2441332351736490543... 1

4

1

4

22

1

42 2

1

21

e e k

k

k

csch

2

( )

=sin2

10

x dx

ex

.24413606414846882031... log9

9 J137

.24433865716447323985... k

kk

1

2

21 2( )

... 2 2 22 1

2 21

1

log( )!!

( )! ( )

k

k k kk

.24483487621925456777... 41

4

1

4 2 12

1

sin( )

kk k k

.2449186624037091293... e

e

1

1 J148

.24497866312686415417... arctan( )

( )

1

4

1

4 2 12 10

k

kk k

6 .24499799839839820585... 39

.2450881595266180682... 63

2

3 3

22 2

1

6

1

6 21

loglog hg

k kk

= ( )( )

11

61

1

kk

k

k

= dxx 1

0

6 )1log(

.24528120346676643... 2 2

20

4

16 42 log

cos

/

Gx dx

x GR 3.857.3

= arctan 2

0

1

x dx

.2454369260617025968... 5

643

0

1 x xarcsin dx

.24572594114998849102...

22

2

1

sin11log21log2)1(

8 k

iiii k

keee

e

i

.245730939615517326... 35

2 .245762562305203038...

0

2

2 )1()!2(

42sinh

2

2cosh

2

1

44

3

2

1

k

k

kk

e

e

.2458370070002374305... 1

2

1 1

2 0

1

sin cos sin

e

x dx

ex

=sin log x

xdx

e

21

... dx

e xx ( log11

)

.2458855407471566264... ( ) log2 2 16 12 21

4 3 21

k kk

= ( )( )

12

41

1

kk

k

k

2 .2459952948352307... root of ( ) ( )1 1

2x

.2460937497671693563...

( )2

221

kk

k

...

4

3

4

12log)2log23(log2

6 22

2

LiLi

8 .2462112512353210996... 68 2 17

.2462727887607290644...

( , , )( )1

44 0

1

4

1

41

k

kk k

.24644094118152729128... 1

14 3 21 k k k kk

1 .24645048028046102679... 2 1 2 2 1 2log( ) arcsinh arctanh 1

2

=1

2

1 2

2 1log

= 1

2 2 1 20 1k

k

k

kkk

H

( )

O

= 1 12

2 31

( )( )!!

( )k

k

k

k !! J141

=( ) ( )!!

( )!!

1 2

2 1

1

1

k

k

k

k k

=dx

x x( )

1 1 2

0

=

1

0 1)1( xx

dx

=

2/

0 sincos

xx

dx

.24656042443351259912...

1

22

2

1

2

)18(

8

8

)2(

22csc

2sin2

32 kkk k

kkk

.2465775113946629846...

( ) ( )2

2 1

2 1

4 111 1

k kk

kk

k

.2466029308397481445... 3

2

3

2 32

1

log( )

1

H

kk

kk

... 1

1728

2

3

1

62 2

2

3 31

( ) ( ) log

x

x xdx

1 .2468083128715153704... e e ee kk

k

log( )( )

11

10

1 .2468502198629158993... arcsec

.2469158097729950883... 1

2 2

1 1e

dx

ekk

x

... dzdydxxyz

xyz

1

0

1

0

1

0

4

1

log

30

.2470062502950185373... log ( )3

2 6 3

1

9 3 321 2

3 21

k k

k dx

x xkk

k x

3 .2472180759044068717... 3 46

4096 30 3870720

14 6

61

( )

( )

( )

a kk

where is the nearest integer toa k k( ) 3 . AMM 101, 6, p. 579

.24734587314487935615... 1

44 4

0

1

( log ) log ( ) sin x x dx GR 6.443.1

.24739755275181306469... 1

232 kk

.2474039592545229296... sin( )

( )!

( )

( )!!

1

4

1

2 1 4

1

4 2 2 42 10 0

k

kk

k

kkk k

.2474724535468611637... 7

4

4

3 23 2

log

.24777401584773147... pq k

kk

( )

2 10

... 33

2

1 12

4 12 21

Gk

k

k

k

( ) ( )

( )

.24792943928640702505... ( ) ( )2 12

k kk

.24803846578347406657... ( )

logk

k kk

k jkk

1

1

1

2

1

42

1

21

12

22

.24805021344239856140... 3

125

.24813666619074161415...

2 6 6

1

2coth

.24832930767207351026...

1

21

21

2

1

4 31

i i

k kk

= ( )( )

Re( )

( )

12 1

4

1

21

11

kk k

kk

k k

i

... 84

1

1 2

3 1

31

( )

( /

k

k k )

.24873117470336123904... 2

2 2

1 2 2 120

1

log arcsin

x

xx dx GR 4.521.3

.2488053033594882285... ( ) ( ) ( ) ( )( )

2 3 4 3 3 51

3

51

k

kk

.24913159214626539868... 7

360 6

4 2 4

40

x

xdx

cosh

1 .249367050523975265... sinh3

... log x

edxx

11

.2494607332457750217... 1

4 11

3

231

12

1k

k

k

kk

k

( )( )

.2498263975004615315... sin sinh( )

( )!

1

2

1

2

1

4 2 2

1

2 11

k

kk k

GR 1.413.1

.250000000000000000000 = 1

4

1

3 6

1

4 421

21

k k k kk k

= 1

1 12 ( ) ( )k k kk

J268, K153

= 1

2 132 1k k

kk k

( ( ) )

= k

kk 4 141

=

3

1

222

)1)()(1()1()1(

12

k

k

k

kkkk

k

= ( ) ( )

( )( )

( ) ( )

1

1

1

1 3

1

2

1

222 0

1

21

1

23

k

k

k

k

k

k

k

kk k k k k k k

= ( )( )

14 1

4 414

1

kk 1k k

k k

k

= ( ) ( ) ( )

1

3 4 1

2

4 1

1

1 1 1

k

kk

kk

k

kk

k k

= dx

x

x dx

x

x dx

x

x dx

x51

3

50

31

2

311

( )

log log

= x x xarcsin arccos0

1

dx

x

x

= x x dlog0

1

= log(sin )sin cos logsin sin cos/ /

x x x dx x x x d0

22

0

2

= log 11

13

x

dx

x

=

44

7

0

3

xx e

dxx

e

dxx

1 .250000000000000000000 = 5

4 22 H ( )

2 .250000000000000000000 = 9

4 3 11

kk

k

68 .25000000000000000000 =

1

5

3)0,5,

3

1(

4

273

kk

k

.25088497033571728323... ( )k

kk

1

2 12

... 12primesmallest;2,lglim 1)(

npnn

n

nppp

Bertrand’s number b such that all floor(2^2^…^b) are prime.

... 1

22 2

1

2 20 00

I Jk kk

( ) ( )( )!( )!

.2517516771329061417... 1

41111

3

2

3

22 2

1

16

1

2

3

21

HypPFQ , , , ,{ , , , },(( )!)

(( )!)

k

kk

2 .2517525890667211077... ( )10

.2518661728632815153...

4 5

5

2

1

10

1

4 521

coth

kk

.251997590741547646964... 3

2

3

3

1

41

1

1

3

( ) ( ) ( )

( )

k

k kk

6 .25218484381226192605... 7 3 4 22

2 2 1

2 1

2

31

( ) log( )

( )

k

k kk

=k k

kk

2

1

2

2

( )

.252235065786835203... 11

2 11

10

1

e e

dx

e ex xlog log( )

.2523132522020160048... 1

5

4 .2523508795026238253... Ik

k

k0 0 1 2

0

2 2 1 22

( ) (; ; )( !)

F

.252504033125214762308...

sec hi i

2

1

2

1

2

...

( )( )

36

2

21

k

kk

.2526802514207865349... arccsc4

1 .2527629684953679957... Li15

7

.25311355311355311355 =691

2730 6 B

... 2 8 3 ( )

2 .25317537822610367757... k

kk

!!

( )!!21

.2531758671875729746... = 11

2

1

2 1

1

1

erf

k k

k

k

( ) ( )

!( )

.253207692986502289614...

7

1 .25322219586794307564... 1

2 3 31

( )

( )f k

kk

Titchmarsh 1.2.15

.2532310965879049271... 9 3 9 3

8 12

1

3 2

2

3 21

log

k kk

... 2/

0

223

cos848

dxxx

1 .25331413731550025121... 2

i i ilog

... 1

42 2

2 21 10

I Jk

k kk

( ) ( )( )!( )!

1 .25349875569995347164... 1

12 kk !

...

0

2

2

3

)3log(

32

12log

x

x

.25358069953485240581... 1

30 ( )!k k!k

.25363533035541760758... 2

32

5

2

1

2

1

21 1

1

120

Fk

k

k

kk

, ,( )

( )!

... 24 81 4

0

1

e e dx /

x

.25375156554939945296... dx

x x

dx

e ex x4 11

3 20

.2537816629587442171... 65

6 453 3 5

1

1

2 4

5 21

( ) ( )( )k kk

2 .2539064884185793236... 2

2

2

21

k

k

k

.25391006493009715683... 1

42

1k

k

.2539169614342191961...

12 2

1

2

11

1log

2

1

kk

kk

Likk

k

.25392774317526456667... 1

2 21 ( )k k

k

.2541161907463435341... 1

44 0

1

4

1

44 41

, ,

Likk

k

.254162992999762569536...

1

sindx

e

xx

.2543330950302498178... 2

4 2 6 CFG D1

.25435288196373948719...

6 3

2

3

3

6 131

2

log log dx

x

.25455829718791131122...

011 )72(!

1,2

11,

2

9F

9

1

k kk

k

.25457360529167341532... 1

32 4 2 7 214 8 2 8 2 7 2 22

( )sinh sinh

i

i i

csch

=1

2 1 21 ( )k kk

.2545892393290283910... 1

2 21

4 4

2 2 4

0

cosh( )!

/ /

16

e e

k

k

kk

1 .2545892393290283910... 1

2 21

1

2 1 4 1621

4

1

cosh( ) ( )!

k kk

k

kk

= sin sinh/

x x d0

2

x

23 .2547075102248651316... 3

4

3

11 .2551974569368714024... 4 13

320

k

k k

k

2 .25525193041276157045... 21

2cosh

.25541281188299534160... arctanh1

4

1

4 2 12 10

kk k( )

AS 4.5.64, J941

4 .25548971297818640064... 7

2

422 ( )

.25555555555555555555 = 23

90

1

2 1 2 70

( )(k k )k

K133

.25567284722879676889... 1

2515 8 5

1

2

1 1

2 1

1

1

( )( ) !( )!

( )!

arcsinhk

k

k k

k

1 .25589486222017979491... 1

20 ( )!k k!k

.25611766843180047273... ( )4 15

6

.2561953953354862697... 4 2 212

622

2

0

1

log log arcsin log

x xdx

.2562113149146646868...

1

8 2 2 2 22 2

2csch sech sinh

=( )

1

2 1

1

21

k

k k

.2564289918675422335... 7

33

1 3

1 3 2

1

e k

k kk

/

( )!

.256510514629433845... log( )

( )

1

1

2

0

1

x

x xdx

60 .25661076956300322279... 333

0

ek

k

k

k

!

1 .2566370614359172954... 2

5

.2566552446666378985... H k

kk

( )3

1 5

2 .2567583341910251478... 4

... csc2

... 1

82 2

2 20 0

2

0

I Jk

k kk

( ) ( )( )!( )!

1 .2570142442930860605... HypPFQ { },{ , },

( )!

11

2

1

2

1

16

1

220

k

k

kk

2 .25702155562579161794... 16 253

6 23

1

log

Hkk

k

1 .25727411566918505938... 5

.2574778574166709171...

2 2 21

2 12

1

log log( )

( )

k

kkk

.25766368334716467709... dxxx

xLi

1

0

2

22

)2)(2(

log

2

1

4

1

8

2log

48

.25769993632568292... ( )

1

2

1

31

k

k k

.25774688694436963...

1

)1)1((1k

k kH

1 .25774688694436963...

( )log log

kk

k k

k

kk km

km22

2 21

=1 1

11

1 11 2 1k

k

k

k

kH k

k

k

kk

k

log ( )

( )( ( ) )

.2577666343834108922... H

kk

kk

( )

( )

3

1 2 2 1

.2578491922439319876... 11 1

3

22 2

1

2

20 1 1 1

0eI I

k

k

k

k

k

kk

( ) ( ) ( , , )( )

!

F

...

1

125

2

52

2

2 30

( ) x dx

e ex x

.25794569478485536661... ( ) ( )

( )4 2 3

2

35

1

1

2

4 21

k kk

... 2 21 1

0

sinh /( )! 2 1

k

k

404 .25826782999151312012... 1

4

2 2

0

e ek k

ke e

k

sinh cosh

!

.25839365571170619449...

( )

( )

k

kk

1 1

2

.25846139579657330529... = Likk

k3 3

1

1

4

1

43 0

1

4

( , , )

=log( / ) log1 4

0

1

x x

xdx

.2586397057283358176... 2

12

226

2 21

2

1

2 1

log( ) ( )

( )

k k k

k kkk k

.258742781782167056... 3

42

7

12 1

2

0

log

xe dx

e

x

x GR 3.42.0

3 .2587558259772099036...

22

2

2

062 2 2 2

log loglog

xdx

ex

.25881904510252076235... sin ( )

12

2

43 1

6 2

4

AS 4.3.46, CFG D1

= 2 3

2

CFG C1

.25916049726579360505... 2 2 21

2

1

2 2 30

arctan( )

( )

k

kk k

.259259259259259259259 =7

27

1

72 0

7

2

1

( , , )k

kk

...

1 21

k

k

kk

1 .2599210498948731647... 2 11

3 23

0

( )k

k k

.2599301927099794910... log8

8 J137

.2602019392137596558... ( )( )

11

21

1

2

1

2 11

2

12

1 1 2

3k

kk k

k k

kk

k

... ( )2 11

kk

.26022951451104364006...

ie e e

k k

ke e

k

i i

4

1

22

2 2 2

1

cos sin(sin )sin cos

!

.26039505099275673875...

1 2!

)1(12logk

kkk

kk

Be Berndt 5.8.5

.260442806300988445...

2

23

41

4

1

10

1

2log loglog

x

dx

x GR 4.325.4

=log x dx

e ex x

0

.2605008067101075324... 11

51

kk

.26051672044433666221... ( ) ( ( ) )

1 1 3

2

k

k

k

.26054765274687368081... 1

2

1

2

1

2 1 41

sinh( )!

k k

k

1 .260591836521356119... cosh( )!

1

2

1

2 20

k kk

.2606009559118338926... 2

2 2112 3 4 3 3

1

2

1

3 1csc cot

( )

kk

J840

.2606482340867767481... 81 3

24

3

2

9 3

281

1

3

2

5 41

log( )

k kk

=( ) ( )

1 4

3

1

1

k

kk

k

.26069875393561620639... ( )k

kk 50

...

0 1

2

),2(!

),2(

k kkSk

kkS

.2609764038171073234... 2

31

3 3

24

2 1

2

( )

( )

k

kk

.26116848088744543358...

21

21

2 1

21

22

1

log(cos ) ( )( )

( )!k

kk

k

B

k k AS 4.3.72

... 21

2 2

1 1

2 22

1

arcsin( )!( )

( )!

k k

k kk

!

1 .26136274645318147699... 1

3 21 k k kk

.26145030431082465654... ( )3 33

21

31

3

i i

4 .2614996683698700833... 742

343 31 3

5

1

ek

k kk

/

!

... arctanh1

2

3

4

1

4 2 11

log( )k

k k k

1 .261758473394486609961...

eG

1 .2617986109848068386... 11

2 2 11

kk k( )

= arccosh x

xdx

dx

e ex41

30

x3

.26179938779914943654... 12

1

6 3

1

4 2 1

2

02 1

0

( )

( )

k

kk

kk k

k

k

=x dx

x

5

120 1

=

07

333 cossindx

x

xxx

... 11 2

122

2

2

2

6

7 3

8

2 2

( log )

loglog log ( )3

= 1 21

2

1

2

1

2 12 3 31

log( )

Li Lik kk

k

1 .26185950714291487420... log3 4

.26199914181620333205... ( )4 3

4 11

kk

k

.2623178579609159738... dxx

xx

1

0

22

1

log

13500

256103)3(16

1 .2624343094110320122... e

e iii

ik

k

/

/

2

201

1

1

.2624491973164381282... | ( )|( ) k k

kk

1

3

1

1

.2625896447527351375...

( ) ( )k kk

k

0

1 2

1 .2626272556789116834... arctan

.2627243708971271954... log

kk

kk

12

2

.262745097806385042621... ( )k

k

k 2 121

.26277559179844674016... 2

228

25

18

4 1

2 2 1

log( )

k

k kk

= ( )( )

11

22

kk

k

kk

...

2 !!

11

k k

... 7 3

32

2

2 20

( )

t dt

e et t

.2631123379580001512... 1

42 2 2 0 1 2 0 1 i i i( , , ) ( , , )

=( )

!( )

1

2

1

21

k

k k k

.263157894736842105 =5

19

...

2

75

1

1

2 2

2 2

p

pp prime

.2632337919139127016... ( )( )

!!

11

2

k

k

k

k

.26360014128128492209... 1

62 2

5

2

1

4 22 1

2 11

Fk

k

kk

k

, , ,

( )

.26375471929963096576... H

kLi

k

kk

kk

( )

( )

2

12

025

1

4

1

5 1

5

4

1

5

.26378417660568080153... 2

3 2118

4

92

16

27

1

2 3

logk kk

5 .263789013914324596712... 8

15

25

, the volume of the unit sphere inR

.263820220500669832561...

1

2

2 )!1(

)1(1

6)(

k

kk

kk

BieLi


.2639435073548419286... 2

2 21

2

1

21

log( )k

k k k

= ( )!

( )!( )

k

k kk

1

2

121 4 1

=2

02

1

0

2 11log)1log(

x

dx

xdxx

= GR 4.591 arcsin logx x d0

1

x

2 .2639435073548419286... 2

2 log

= log log( )

11

12

220

1

0

xdx

x xdx

...

2 2

)()1(

kk

k kk

1 .2641811503891615965... k

kk ( !)31

.26424111765711535681... 12 1

2

1

1

1

10

1

1

4

0

e k k

k

k

k

k

k

k

k

k

k

k

( )

!( )

( )

( )!

( )

( )!

=1

2 1

1

11 0( )!( )

( )

( )!k k k k!k

k

k

= ( )! !

( )log

!

11

12

1 1 2

k k

k n

kn

k

k H

k n

k

k

= xe dxx

xdxx

e

log2

10

1

.264247851441806613... ( ) log ( )( )

3 116

3 1

4

3 1

21

2 3

4 1

2 1

0

kk

k k

.264261279935529928...

2

25

7

5 1

2

5 20

csc

( )

x

x

1 .2644997803484442092... 1

2 1

1

2 10 1k

k

k

kk

k

( ) ( )

.2645364561314071182... 9121

5

1

1 60

ek k kk

!( )( )

... 3

8 2

1

4 2 1

21 2

1

( )

( )

k

kk

k

k

k

k

4 .26531129233226693864... ( )k

Fkk

12

...

1 4

sin

1cos817

1sin4

kk

k

1 .26551212348464539649... log log21

2

4744 .2655508035287003564... 8

2

1 .2656250596046447754... 1

20k

k!

.2656288146972656805...

( )

( )

3

4 1

1

413 3

1

kk

k k

k

1 .2656974916168336867... 1

21 4 4 1 2 2 1 12 2

ee e e e si c( ) cosh ( )( ( ) ( i ))

2

( ) log ( ) ( ) ( ) ( )1 2 1 2 12 2 2e e ci e si =

H

kk

k ( )2 11 !

.26580222883407969212... sech21

2

22 2 cosh e e

.265827997639478656858... F kkk

( )2 1 12

.2658649582793069827... 2

3 122 3

G

log( ) Berndt 9.18

.2658700952308663684... 1

2 11

k kk

( )

4 .2660590852787117341... 1 1

11e e

ek

e

e

kk

( , , )

... Io ( )1

=1 21

22

0e

k

k

k

k( )!

( !)

= e dxcos

0

1

x

... 9

4

1

4

5

6

1

3

1

2 31 1

21

( ) ( ) ( )

( /

k

k k )

118 .2661309556922124918... 31

252

465 6

4 1

65

0

1

( )

(log )xdx

x GR 4.264.1

x dx

ex

5

0 1

.2662553420414154886... cos( ) 2

1 .2663099938221493948... ( )

1 2

1

k

kk

k

F

5 .26636677634645847824... 2

2

( )kk

2 .26653450769984883507... dx

x( )1

... 1

2 13 2

2

0

cothsin

cschx x

e ex x

.26666285196940009798... | ( )| 2

41

kk

k

1 .26677747056299951115... 8 2 2 2 21 2 1

0

log log( )

(k +1)8k

k

k

k

.26693825063518343798... dxex

x

e

e

1

02)1(

log

1

)1log(1

... 11

8 2

3

1

e k

k kk

!

.2673999983697851853... 1

22 2log

= log cos sin log 8 8

1

22 2 2 2

=1

41 k

k

k

cos

=

1

4 41 2 1 1 1 11 4 1 4 3 4i

( ) log( ( ) ) log( ( ) )/ / /

...

4

32log2log23log2

64

12

2

2 LiLi

=( ) log( / )

1

3

1 41

1 0

1kk

kk

H

k

x

xdx

5 .267778605597073091897... 2 2 7 31

22

0

log ( )sin

cos

x x d

x

x

... ( )

( )! (

1 2

1 21

k k

k k k ) Titchmarsh 14.32.1

.26794919243112270647... 2 31

6 1

2

1

kk k

k

k( )

.26794919243112270647...

k

k

kkk

2

)1(6

133

0

4 .2681148649088081629... 160

27

8

27

64

272

35 2

22

0

log ( )/x Li x dx

.268253843893107859...

6 4 23 3

21216 12 2

1 2 1

( ( ) )( )f k

kk

Titchmarsh 1.2.14

.268941421369995120749... 1

1

1 1

1e e

k

kk

( )

.26903975345638420957... erfi

ee x xx1

4

2

0

sin cos dx

... ( )

11

k

kk

k

F

.2696105027080089818... ( ) cos cos(log )

1

1 1 1

1

21 0 0

1k

kx

k

k

x

edx

x

xdx

=1

41

21

2

1

2 2

1

2 2

i i i i

=

2

1

2

1

224

1 iiii

4 .2698671113367835... e e x xdx

xx

cos sin(sin )0

GR 3.973.4

.270034797849637221...

2 2 22 20e

x dx

x

cos

.27008820585226910892... 16

105

.27015115293406985870... cos ( )

( )!

( )

( )!( )

1

2

1

2 2

1

2 1 2 10 1

k

k

k

kk

k

k k

.27020975013529207772... 8

1

2

1

4 141

2

arctan

x dx

x

.27027668478811225897... 2

2116

2

2 1

log log

( ) ( )

x

x xdx

1

.270310072072109588... 2

3

3

21 1

1

log ( )

k k

k

H

2k J 137

.2703628454614781700... log( )

( ( ) )2 11

12

k

k kk

=1 1

1 12 k kk

log/

K 124h

1 .2703628454614781700... log 2

=1

1 20

xx

dx

xcos

.2704365764717983238... 14 2 2

1

4 421

1

tan

k kk

4 .27050983124842272307... 5

23 5 5

1

5

2 2

0

( )

kk

k

k

... dxx

x

1

061

1log

32

32log

2

32log5

.270580808427784548...

1

3

4

)1,3()1(4

)4(

360 k

k MHSk

H

.270670566473225383788...

1

3

1

2

1

1

2 !

2)1(

!

2)1(

)!1(

2)1(2

k

kk

k

kk

k

kk

k

k

k

k

ke

3 .27072717208067888495... 2 212

2

3

3 3

2

3 3 4

40

2

loglog ( )

sin

/

x

xdx

1 .27074704126839914207... ( )

!

1

20

kk

kk

B

k

...

1

2

4

5,1

24

2 kk

kk

G

.27101495139941834789... cosh

1

2

1

2i i

=

1

21 )12(2)1(k

kk k

.27103467023440152719... ( )3 1

6 2 3 2 3 1 1 1

4

120

6

120

x

xdx

x

xdx

.2711198115330838692... ( )

1

2

1

31

k

k k k

... e

e

k

k kk16

1

16

1

4

1

2 2 11

sinh( )! 4

1 .27128710490414662707... 1

4 1

1

4

5

41

01k kF

k !( ), ,

1

.271360410318151750467... 1

2 2

1

2 2 21

sinh( )!

k

k kk

... cosh

sinh1 1

2

12

13

x

dx

x

... 212

2 2 22

2 2

0

2

log log logsin sin/

x x dx

1 .2717234563121371107... 0 10

21

22 2

1

2 1F1 ; ; ( )

!( )!

Ik kk

k

.2717962498229888776... 63 5 3 45 3

150

1

3 51

log

( )k kk

.2718281828549045235... e

10

1 .27201964595140689642...

.27202905498213316295... sinh

11

11

2

121

22k kk k

=

( ) ( ) ( ) ( )i ii i

i i

3 3

101 1

= )2()2(10)2()2(2

1iiii

1 .2721655269759086776... 6

9

.27219826128795026631...

2

1

2

1

28

2log22

iLi

iLi

iG

= dxx

x

1

021

)1log( GR 4.291.8

= log( )x

xdx

1

1 21

GR 4.291.11

=

log x dx

x1 40

1

GR 4.243

=x dx

e ex x 2 20

2log

GR 3.418.3

=

log(sin )sin

sin

/

xx dx

x1 20

2

GR 4.386.1

= log1

1 20

1

x

x

dx

x GR 4.297.3

=x dx

x x(cos sin )cos

/

0

4

x

= x x xarccos log0

1

dx

=arctan x

xdx

10

1

= log( GR 4.227.9 tan )/

10

4

x dx

= log((cot GR 4.227.14 ) )/

x d 10

4

x

.272204279827698971... 1

1242 kk

.2725887222397812377... 4 25

2

1

2 21

log( )

kk k

6 .2726176910987667721... 8 21

4

12 2 1

161

cotkk

.27270887912132973503... 3 2 21

2 11

2

e Eik

k kkk

log ( )( )!

.27272727272727272727 = 3

11

.27292258601186271514...

( )( )( )

12

1

11

kk

1 .27312531205531787229...

1

22

24

2

1)2(2

5tan

521

13 kk

k

kFkk

k

.273167869100517867... 2

7

1

2 2

1

221

arcsin

kk k

kk

1 .2732395447351626862... 4 2

3 2

1

1 2

5

4

2 3

22

2

2

( )

( / ) /

( )!!

( )!!

k

kk

J274

= 1

2 202k

kk

tan

K ex. 105

= ( )

( )

2 1

2 2 2

2

1

k

k kk

.2733618190819584304... 1

2 121

kk

.27351246536656649216...

1

27

2

3 12

2

31

2

21

( ) log x

xdx

x

e edxx x

9 .2736184954957037525... 86

1 .27380620491960053093... log log( )3

21 3

12

20

1

xdx

x GR 4.295.38

.2738423195924228646... 1

28 2 14 2 13

2 32

1

log log( )

H

kk

kk

.27389166654145381... 1

254 3 3 27 3 2 32 log ( )

=1

31

3

34 31

1

1k k

k

k

kk

k

( )( )

.2738982192085186132... dx

x x4 41

... ( )k

k k k kk k

1

2 1

1 1

22

2

arctanh1

.27413352840916617145...

2 3

31

3

2 21

sin

( )

kk

.2741556778080377394...

1

2

23

cos

36 k k

k

= dxx

x

x

dx

x

1

0

6

13

)1log(11log

2 .2741699796952078083... 32 5 2 71

8

2 3

0

( )

kk

k

k

1 .2743205342359344929... 2 2

11

1 221

coth( /

)

kk

J274

28 .2743338230813914616... 9

...

1

112 2

)1(4

1,

2

3,2,2

2

1

k

k

k

kk

F

.2745343040797586252... 83711

304920

1

111

k kk ( )

.27454031009933117475... k

Likk 5

1

5

1

5

1

20

11 2

/ ( , , )

.2746530721670274228... log

( )

3

4

1

2

1

2

1

4 2 11

arctanhk

k k

=dx

e x20 2

.2747164641260063488... ( ) arctan log

14

1

52 2 2

5

41 2

1

k kk

k

H

7452 .274833361552916627... 9

4

.27489603948279800810... 4

9 62

1

4 31

log

( )k kk

1 .27498151558114209935... 1

3 21k k

k

.275000000000000000000 = 40

11

.27509195393450010741... sin

( )( ) ( )

k

k

ie e e e

k

i i i i

1 221

22

2Li Li

3 .27523219111171830436... 3

G

.27538770702495781532... 2 2

2122

2

21 2

1

21

log

loglog Li

=1

2 121

kk k k( )

... ( )

! ( )

1

2

1

1

k

k k k

.27543695159824347151... ( )

1

13 30

k

k k k k

=1

42

2

2

1

81

21

2

1

2 2

1

2 2

csch

log( ) ( ) ( ) ( )

i i i i

.27557534443399966272... 1 1 1 2

2 11

1

1k k

k

kk

k

k

arctan( ) ( )

( )

1

x

... log tanx x d0

1

.27572056477178320776... cschB

21 1

2

2 2 1

22 2

2 2 12

1

e e k

k kk

k

( )

( )! J132, AS 4.5.65

.27574430680044962269... loglog(cos ) log( cos )

( )cos

21

2

2

81 1

1

4

k

k

k

k

... 4/

0

23

2

cot)(2

1)1log(

162

)3(

4

dxxxiLii

G

... Levy constant

... 8

5

2

5 5

4

5

4

5 1 5 20

csc /

x dx

x 4

.27591672059822730077... ( )( )

( )

( )

1 11

12 2

k

k kk

k

k k

... 1

4 2 22

1

2 3

1

21

csch

( )k

k k k

.27614994701951815422... 1

8 12 3

1

9 421

kk

.27625744117189995523... 5

324 3

3 2

3 30

log x

x xdx

1 .27627201552085350313... 13

32

3 3

50

x x

xdx

sin GR 3.787.3

.276326390168236933... Erfk k

k

kk

1

4

2 1

4 22 10

( )

!( )( )1

.2763586311608143417...

( ) ( )

( )

3 1

6

1 13 3 3

1

k kk

... log( )

25

12

2 1

4 21

k

k k kkk

.27657738541362436498... ( )

!

1

1

1

1

k

k k

2 .27660348762875491939... 2

2 3 222 0 1

0eI e e

e

k k

k

k

( ) (; ; )!( )!

F

...

0

2/)1(

02/3 2

)1(

2

1

2)14(

1

4

1,

2

3

8

1

kkk

kk

k

[Ramanujan] Berndt IV p. 405

... ( )

13

1

1

k

k k

k

.27700075399818548102... 43

234

9 13

23413

csc

= ( )

1

1324

k

k k

.2770211831173581170...

2 1 12

12

1 1

1

( ) ( )! ( )

( )!

k k

kk

k

k

...

0

2sinlog565log32arctan8100

1

25

14dxxxex x

.27708998545725868233... 1

4 12k

k k( ( ) )

.2771173658778438152... ( ( ) ) kk

1 3

2

4 .27714550058094978113... 1

22

1

2, ,

.27723885095508739363... H

k

H

kk

kk

kk

kk4

1

321

1

1

( ) (2)

.27732500588274005705... log2 2 .27734304784012952697... ( )1

3

... x x

edxx

log( )1

10

1 .277409057559636731195...

12 23

1

4

3log3

12

3

k kk

= dxx

x

1

03

3)1log(

.27750463411224827642...

1

2 )!1(

)1(1)1(

k

kk

k

BeLi

1 .27750463411224827642...

0

2 )!1(

)1()1(

k

kk

k

BeLi

.2776801836348978904...

8 2 2 12 20

2

4 20

dx

x

x dx

x( ) ( )

192 .27783871506088740604... 6

5

1 .27792255262726960230... 2 2 2 2sin log ( ) i i i

.2779871641507590436... log7

7

... ( )

( )

1

2 1

1

21

kk

k

H

k k

.27818226241059482875... 12

2

1

2 1

21

1

arcsinh ( )k

k k

k

k

... ( )

!( )!

2

1

1 2

1 11

k

k k kI

kk k

... 1

164 3

1

2 1

2

20

1

, ,log

/

x

x 4

=

3

23 34

2 22 2

log i Li

iLi

i

.27865247955551829632... 11

2 2

1

2 21 1

log kk

x

dx

17 .27875959474386281154... 11

2

.2788055852806619765...

1

1,2

1)11(

xe

dxerf

x

... e

e x

dx

x2

5

22

1

14

sinh

.27892943914276219471... 5

4

5

4 51

log

Hkk

k

... k kk

log ( )

2

.27918783603518461481... ( ) ( )

logk k

k

k

kk

kk k

1 11 1

1

22

1

.27930095364867322411... 33

2

1

6 1 6

1

3 2 320 0

k kk

k

kk/

( )

( )

.27937855536096096724... S k k

k kk

22

1

2

2

( , )

( )

.27937884849256930308... 1

2

1

21

1

2

3

2

1

2 2

1

1

, ,( )

k

k k

=1

22 2 2

1

2

( )

.2794002624059601442... 1

4 1

1

4 11

1

1k

k

k

kk

( )

.27956707220948995874... 1

22

1

2 2

20

1 2

0

Jk

k

k

k

k

kk

( )( )

( )!

2 .2795853023360672674... Ik

k

kk k0 2

0

2

20

21

( )( !) ( !)

LY 6.112

= 0 12

0

2

111

2F e(; ; ) cos

d Marsden p. 203

= 1

2

2

2

2 1 2

0

2

0

2

0( )! ( )!

( )

!k

k

k

k

k

k

ke

k

k

kk k

k

k

.2797954925972408684... 1

102 1 5 1 5

1

531

( ) ( )i ik kk

1 .2798830013730224939... ei

e ek

ke e

k

i icos sin(sin )sin

!1

1

12

GR 1.449.1

1 .27997614913081785... 2 2 1

22

1

1

erfi( ) ( )

!e

kk

k

k k

k

8

=

7

25

1

1

6

2 3

p

pp prime

.28010112968305596106...

41

2

0

2

ee

x

edx

x( )

sin

7 .2801098892805182711... 53

.28015988490015307012... ( )k

k

kk 21

.2802481474936587141...

3

1

61

3

21

3

2

1

2 331

i ik kk

.2802730522345168582... 1

63 9 3 2

2

3

1

3 11

21

log ( )( )

( )

k kk

.28037230554677604783... 5

32 2

3

2

1

2 22

log hgk kk

=( ) ( ( ) )

1 1

2 12

k

kk

k

1 .28037230554677604783... 8

32 2

3

2

13

22

log( )k kk

.28043325348408594672... 1

36

1

3

1

6 41

21

( )

( )

kk

1 .28049886910873569104... 1

2111

1

2

1

2

1

1620 k

kk

HypPFQ { , , },{ , },

... ii eei

2log2log2

1sin)1(

=

2

1)(sin

k

kk

k

6 .28063130354983230561... k kk

( ( ) )2

2

1

.2806438353212647928... 2 2 21

22 log log

l Berndt 8.17.8

.28086951303729453745... 352

735

2 2

7

1

2 71

log

( )k kk

...

1 2!

)1()21(

2

1log

kk

k

k

ke

=

1 2!

)12()1(

kk

kkk

kk


.2809462377984494453...

64

27

1

2

3

428 3

64

27

12 33

43

1

( ) ( )( )kk

.2810251833787232758... 2

3 2124

25

192

1

4

k kk

... 2 31

22

1 22

2 1

1

1

log3

2

( )

( )

k k

k k

kk k

... 33 1

3 1log

.28128147005811129632... 1

22

1

2

1 1

1

log( cos )( ) cos

k

k

k

k GR 1.441.4

.28171817154095476464... 320

ek

kk ( )!

=k

k k k k kk k!( ) !( )( )2 6

1

2 30 0

1 .28171817154095476464... 42

2

0

ek

k kk !( )

.28173321065145428066... e

e

e

e

k

k

k

k

1

2

2

2

12

2

1 2

1

cos(sin ) ( ) sin

!

cos

cos

.28184380823877678730... 4

3

8

7

82 2

8

3

8cot cot log log sin log sin

=( )

1

4

1

21

k

k k k

.2819136638478887003... log( )

224

3 1

2 2 1

2

2 21

k

k kk

=( ) ( )

1

2

1

3

k

kk

k k

.28209479177387814347... 1

4

.28230880383984003308... ( )

1

3

1

1

kk

kk

H

k

(3)

.282352941176470588 =24

85

4

0

sin x

ex

1 .28251500934298169528... 3 6 21

1 230

log( )

( )(

k

)k k k

...

2 3

22

2

3

8 2

32 2 3 4 2

1

43 3

log loglog log log ( )

Li

4

1

43Li

=log ( )

( )

2

120

1 1

x

x xdx

1 .2825498301618640955... 6

2 ( )

.2825795519962425136... 1

21

41 20

Ik

k kk

( )( !)

.28267861238222538196... 1

631 kk

.2827674723312838187... 1

332 kk

.2829799868805425028... 0 11

0

2 22 2

2

1 2

1F (; ; )

( ) ( )

!( )!

J

k k

k k

k

.2831421008321609417...

( )

( )

k

k kk

1

2

.28318530717958648... 2 6 2

0

1

arcsin arccosx x dx

6 .28318530717958648... 21

6

5

6

=1

14

340 ( )( )k kk

= log 1 6

0

x dx

=d e

e

x

x

2 10

2 6

cos

/ dx

6 .2832291305689209135... 2 2 coth

2 .28333333333333333333 = 137

60 5 H

.2833796840775488247... ( )

( )!sinh

2

2 1

11

1 1

k

kk

kk k

.2834686894262107496... 11

31 3

1

1

ek

k

kk

/ ( )

!

5 .28350800118212351862... 2 1 23 4 4

0

/ log

x dx

.28360467567550685407... 3 2 3 33

2 3 1

1

2 11 0

log log( )

( )

( )(

3)

k

k k kk

k

kk k

.2837571104739336568... log( )

log

2 2

11

2

1

22 1

k

k kkk k k

.28375717363054911029... 6

251

6

5 61

logk Hk

kk

.28382295573711532536...

2

24

1

6

49

36

14 2 4 1 2 4

kk

( )( ) ( , ) ( , , )

.2838338208091531730... 1

41

1 1 2

2

1

120 0

1

0

1

e k

x

edx

xdx

k k

kx

( )

( )!

sinh

coth

1 .28402541668774148073... e4

.28403142497970661726... ( ) ( )

( )!

2 2

21

k k

k

2

k

=1

1 11 1

2 22 k kk

coshk

.28422698551241120133... ci ci( )sin ( )sin ( )cos ( )cos( )1 1 2 1 1 1 2 1 si si

=sin x dx

x10

1

6 .2842701641260997176... k Hk

kk

2

1 2

... 2

22 2

2122

3

6

2 3

4

1

22

log

log log loglogLi

1

2

1

2

21

4833Li ( )

=log ( )

( )

2

20

1 1

2

x

x xdx

2 .28438013687073247692... ( ) ( )3 4

... ( )

( )!sinh

k

k k kk k2 1

1 1

2 1

k

1

2 .28479465715621326434... 8

11

3 .28492699492757736265... F

kk

k !!

1

... 2 2

24

log

... 1

21 k kk ! ( )

...

0

24/9

sin2

22 2

dxxe x

.28539816339744830962... 4

1

2

1

4 1

1

21

( )k

k k J366, J606

= sin cos sin cosk k

k

k k

kk k

1

2

1

= x x dxarctan0

1

.2855993321445266580... 11

1 .2856908396267850266... 15

32

1

32 2

4

0

e

e

k

kk

( )!

.285714285714285714 =2

7

.285816417082407503426...

13

2

)3(

1

162

11

543

)3(

k kk

1 .2858945918269595525... ( )!!

1 12

1

k

k

k

k

.28602878170071023341... 2

21

2

16

3 2

4 2

8 1

4 4 1

log

( )

G k

k kk

=k k

kk

( )

42

.2860913089823752704... ( )3 G

.28623351671205660912... 2 1

3 2124

1

8

1

2

( )k

k k k

.2866116523516815594... 6 2

16

CFG D1

.2868382595494097246... 1

4HypPFQ { , , , },{ , , },

( !)

( )!( )1111

1

22 2

1

4 2 2 2

2

20

k

k kk

...

4 2 2 8 2

2

9

2

2 1

22

2

22

coth( )

cschk

kk

= ( )( )

12 1

21

1

kk

k

kk

1 .28701743034606835519... e erfk k

k

1 32

1

12

1

4 2

1

4/

!!

1 .287044548647559922... 11

22

( !)kk

... e k

k

( )

1

2

1

1 .2872818803541853045... 2 17

18

3

3

3 2

1

e k

k

k

k

( )!

.28735299049400502... Patterson Ex. 4.4.2c4

1

12010 2 20 3 15 2 30 4 24 55 3 2 2( ( ) ( ) ( ) ( ) ( ))

2 .2873552871788423912... e

k

k

k

k

sinsin

!

/

12

42

1

...

2

1

2

1

4

2log2 22

iLi

iLiiG

... 2 11e ee ei icos cos(sin )

e

3 .2876820724517809274... 2 2 31

41 1

1

41

log log ( , , )

kk k

J117

=dx

x x

dx

x x

dx

e

dx

ex x21

2

21 00

2

3 1

log

3 1

=( )

( )

1

3

1

42

1

7 2 12

1

7

1

11 2 1

0

k

kk

kkk k

Li arctanh

10 .287788753390262846... k

k

e

k !

0

1 .28802252469807745737... log1

4

2 .288037795340032418... ( )

.2881757683093445651... 6

5

1

5

1

51

1

521

1

1

hgk k

k

k

kk

k

( )( )

3 .28836238184562492552...

1

3

!

)(

k k

kt

.28852888971289910742... i i i

kk6

2

3 3

2

3 3

1

3 2 20

( ) 1

.2886078324507664303... 2

2

0

( )e edx

xx x GR 3.463

( )ex

dx

xx 2 1

120

GR 3.467

.2886294361119890619... log

( )(

2

5

3

20

1

2 1 2 60

)

k kk

.2887880950866024213... 11

21

kk

=

12/)3(2/)3( 22

2

1

2

1)1(1

kkkkk

k Hall Thm. 4.1.3

.28893183744773042948...

42

0

2

ex x x d sin(tan ) sin tan

/

x GR 3.716.8

.2889466641286552456... 1

4 2 11k

k k( )

.2890254822222362424...

201

sin x

edxx GR 3.463

.289025491920818... 1

one ninth constant

1 .28903527533251028409... 4

3

1

3 2 1

3

2 2 120

2

20

( )

( )

( !)

( )!( )

k

kk

k

kk

k

k k

=

3 3

310

275

1

3 1

2

20

log( )kk

Berndt 32.7

=

3 33

10

27

5

9

1

3

21log ( )

=

4

3,2,2,

2

1,1,1,1,1HypPFQ=

2

1,2,

3

1

3

1

5 .28903989659218829555...

1

5

... ( )

log

12

2

k

k k k

.289144648570671583112... ( )

1 1

1

k

kk F

.28922492052927723132... 2 3 3

48

1

3 3 10

( )

( )(

k

k k k )

2 .2894597716988003483... 2 log 2 .2898200986307827102... li( )

...

1001

3455

1

1

2 4

2 2

p p

pp prime

.2898681336964528729... 2

2 2113

31

1 1

22

k k

k

k kkk ( ) ( )( ) K Ex. 111

= k kk

( ( ) )

3 11

1 .2898681336964528729...

2

232 2 2 2 1 1 1

( ) ( ( ) ( ) ( ) )k k k kk

= 11

12 21

k kk ( )

2 .2898681336964528729...

1

2

2)2()1(13 k

kkk

= dxe

ex

x

0 1

1 GR 3.411.25

=

1

10

1 x

xx dxlog GR 4.231.4

3 .2898681336964528729... 2

233

2 21

2

( )( ) (k k

kk

)

=x dx

x

x dx

x

x dx

x

2

2

2

20

2

211 1sinh

log

( )

log

( )

GR 4.231.4

=log ( ) log( )2

20

1

0

11 1

x

xdx

x

xdx

=x dx

e ex x

2

0 2

=

0 13/1xe

dx

.28989794855663561964... 6 1

5

CFG C1

.2899379892228521445... 11

1 11

1

1

1

e

eH

e

kk

kk

( log( ))( )

( )

57 .289961630759424687... cot cot1180

.290000000000000000000 =29

100

1

7 621

k kk

... ( )

1

2

1

1

k Ok

kk

H

... 2 21 1

0

/ (e k

k

k

) / !

.29051419476144759919... sin x dx

ex

20

=1

41 1

1

211

1

22 22 1 2 1F i i F i i, , , , , , ( ) cos(log( ))

csch

2 .29069825230323823095... e erfkk

( )( )

11

120

!

.2907302564411782374...

4 3

3

2

1

6

1

4 321

coth

kk

.29081912799355107029... 4 81

2

1

4 2 30

arctan( )

( )

k

kk k

... 1

2 2

1 1

1 10

1

e

e x

e xdx

log( ( ) )

( )

.291120263232506253... 2 2 2

20

1

24

2

4

1

1

log log( )

( )

x dx

x x GR 4.295.17

=

log( / )1 22

0

1 x

xdx

.29112157403119441224... 1

5 1

3

531 1k

k

kk

k

( )

.2912006131960343334... 2 2 2 2 2 12

2 1 11

1

1

cos sin ( )( )!(

kk

k

k

k k )

.2912758840711328645... 1 1 2

12

1

2

1

42 2

1

1 1/( )coth

kk k

dx

x

1 .291281950124925073115... 3

24

1 .29128599706266354041... 1

1 0

1

k

dx

xkk

x

GR 3.486

5 .2915026221291811810... 28 2 7

.29156090403081878014... G

.29162205063154922281... 1

4 16

2

4

7 3

8 1

2 2 3 2

20

1

log ( ) arccosx x

xdx

.29166666666666666666 = 7

24

1

2 1

1

214

0

1

( )

( ) (

k

kk k

dx

x )

.29171761150604061565...

( )2

31

kk

k

.29183340144492820645... 3 4 13

314

2

k

k k

kk

( ( ) )

.2918559274883350395... ( )k

kLi

k kk k3

23

1

1 1

.2919265817264288065... cos ( )( )

2 12 1

1

1 1 12

2

!

kk

k k GR 1.412.2

= ( )

( )!( )

1 4

2 1

1 2

1

k k

k

k

k k

... 3 2

2124 1

arctan x dx

x

.291935813178557193... ( )( )

!/

11

11

2

1

2

k

k

k

k

k

ke

k

.292017202911390492473... 3

4 2 0

1

G

x x xarccot log dx

.29215558535053869628... ( )

!( ( ) )

1

12

k

k k k

.29215635618824943767... 1

2 6 2

12

3 2

2 21

coth

( )

/

k kk

.29226465556771149906... ( ) ( )2 1 2 1 11

k kk

.292453634344560827916...

( )log

k

kk

1

1

1

23 2

2

=

kk kk

log 11

11

22

.2924930746750417626...

2 5 5

1

2

1

5 121

coth

kk

.29251457658160771495... 1

844

2 2

1

e ek k

kk

cos cos(sin )sin cos

!

.29265193313390600105... H

kk

kk

( )3

1 4

.2928932188134524756... 11

2

= ( ) ( )( )!

( !)

11

4

21

2

41

1

12

1

kk

k

kk

k

k

k

k

k

.2928968253968253 = 7381

25200 10

1

10

1

2510

21

26

H

k k kk k

3 .293143919512913722... 1

2

1

2

3

21

2

3 2

1

( , , )/

kk

k

3020 .2932277767920675142... 7

.29336724568719276586... log( )1

1 30

1

x

xdx

.29380029841095036422... sinh

cosh1

4

14 5

1

x

dx

x

1 .2939358818836499924... ( ) log ( )k kk

2

.2941176470588235 =5

17

.2941592653589935936... 10

51

50

1

2521

coth

kk

J124

.29423354275931886558... i

i ix x

e ex x22 2

11 1

0

( ) ( )( ) ( )sin

... ( ) ( )

log

1 1 11

1 12

2 2

k

k k

k

k k

k

k k

k

.29430074446347607915... ( )

1

2 11

k

kk

.2943594734442134266... Ik

k k kk

20

21 2

( )!( )!

.2943720976723057236... k k

kk k

k k

6 3

3 32

2

113 1

( )( ( ) )

.29452431127404311611... 3

32

33

22

1

k k

kk

( )

( ) J1061

.2945800187144293609... 412

2 21

2

2 1

3 21

log

( )k

k k k

.29474387140453689637... 1

4

2 1

12

( )

( )

k

kk

.29478885989463223571...

22

2

)(

111

)(

)1)((

kk kk

k

= 112

( )

( )

k

k kk

2 .2948565916733137942... ( )kk

2

... e2

4 .29507862687843037097... k

kkk

3

1

.2954089751509193379...

6

... 2 2

31

14

1

log log( )

x

xdx

= dxx

x

1

0

2 11log

3 .295497493360578095... e

.2955013479145809011... 11

12

22

12

2

( )( )k k

H

k

.2955231941257974048...

( )( )

11

k kk

57 .2955779130823208768... 180

, the number of degrees in one radian

.29559205186903602932... ( )( )k k

kk

1

41

... ( )

1

4 4

1

21

k

k k k 5

.29583686600432907419... 3 3 3 31 3

0

log

ee

x

dx

xx

x

GR 3.437

3 .29583686600432907419... 3 33

1

1 2 3 1 3 3 1

0

1

log/ /

x x x x

xdx

n n n n

GR 3.272.2

...

log log

( )( )2 2 2

21

2

2 12

2

123

1

2

k

kkk

.29593742160765668... 10 14 22 1

2

1

2

log( )(

k

k kk )k

5 .2959766377607603139... cosh

sinh( )!

1

2 2

2

4 42

4

0

e e

k

k

k

.29629600000000000000 = 37037

125000, mil/mile

.2965501589414455374... 9 3

121

13

0

1

3

x

xdxlog

.29667513474359103467... 11

21

k kk

.29697406928362356143... 7 312

4

12

41

( )

( )

k

kk

.29699707514508096216... 1

2

3

21

2

121

0

e

k

kk

k

k

( )( )!

.297015540604756039... 1

2

3

12

3

2

1

4 421

2

tan

k kk

.2970968449824711858... 2 2 23

21

1 2

2 12 1

1

1

Fk k

k

k

k

, , ,( ) ( )!!

( )!!

.2974425414002562937... 2 21

1 20

ek k

k

( )!

.2974425414002562937... 22 1

2 20 0

ek

k

pf kk

kk

k

!

( )

.29756636056624138494...

0

1

246

1

612

)1(

k

k

k

.2975868359824348472... 2

22

206

31 5

2

1

2 2 1

log( ) ( !)

( )!( )

k

k

k

k k Berndt 32.3

.29765983718974973707...

0

3

3

3

4

3

1xe

dxx

...

1

1

2)1(

1k

k

k

k

.2978447548942876738... Hk

kk

3

1 6

.29790266899808726126... 1

0

2arctan)223log(224

1dxx

.2979053513880541819... H

kk

kk

(2)

41

.29817368116159703717...

e

Eixe dx

x

x

21

1

2

20

( )

.29827840441917252967... 1 2

22 k

k

kk

log

.2984308781230878565... 1

21

logk

kk

... 2

3 29

2

2

20

1

log( )log( )

x

xdx

... ( ) ( )

1

1

1

2 12

k

kk

k

j kjkk k

3 .2985089027387068694... 32

945

49

, volume of the unit sphere in R

.2986265782046758335... log6

6

.29863201236633127840...

0

2

)!3(

2

8

5

8 k

k

k

e

.2986798531646551388...

3 33

3 3 13 230

log( )

x dx

e x GR 3.456.2

=

x x dx

x

log

( )1 3 230

1

GR 4.244.3

.29868312621868806528... ( )( )

12 1

11

1

k

k

k

k

.2987954712201816344... 15

1624 3

3 3

8

1

3 41

log

( )k kk

1 .29895306805743878110...

0 22

1

22

1arcsin

77

8

7

8

k k

k

k

1 .2993615699382227606... ( )k

kk2

2 2

.29942803519306324920... ( ) cot2

1

2 2 2 2

1

2

2

4 21

k

k kk

= ( ) ( )2 2

21

2k kk

k

.29944356942685078... ( )

1

532

k

k k

.2994647090064681986... 6

3 3e e J132

8 .29946505124451516171... 1

211 1

0

e e ek

ke e

k

/ cosh cosh(sinh )cosh

! GR 1.471.2

2 .2998054391128603133... 3 1 12

120

logx

dx

.299875433839200776952... 2 2

4148 3

1

3

2

4

G x

xdx

log arctan

5 .29991625085634987194... ( , )1

3

1

3

1

32

3

2

.300000000000000000000 = 3

10

1

2 31

0

2 1 3

cosh log( ) ( ) logk

k

ke J943

= sin cos3

03

0

x

edx

x

edxx x

.300428331753546243129...

1

0

12

)()(

kk

kk

.300462606288665774427... 12

13

.30051422578989857135...

( )

( ) log log3

42 3

2

32 2

1

22 2

1

23

2 3

Li Li


= log log

( ) ( )

2

31

2

1 1 1

x

x xdx

x

x x xdx

= log2

1

2

1

x

xdx


= x dx

e x

2

20 1

= log ( ) log( ) log2

0

1 2

0

11 1

x

xdx

x x

xdx

= logsin tan/

x x2

0

2

dx

2 .300674528725270487137... 2 3 ( )

.300827687346536407216... H

kk

kk 2 2 11 ( )

1 .3010141145324885742... ( ) ( )

( ) ( ) ( )3 4

3

90

41

41

13

1

k

k

k

kk k

HW Thm. 290

... log10 2

... 0

1

( )

!

k

kk

.30116867893975678925... sin cos1 1

=

1 1

11

)!2(

)12()1(

)12()!12(

)1(

k k

kk

k

k

kk

= 3

1

1

0

1sinsin

x

dx

xdxxx

= dxx

xxe

1

logsinlog

... 2 11

11

1

1

4

20

20

log logx

xdx

xdx

=

022 )1)(2/1( xx

dx

2 .3012989023072948735... sinh Re{sin(log )} cos ( ) 2 2

1

i i i

= 11

2 1 20

( )kk

.31034129822300065748... 2

12sin

1

)1(

2

1cos

2

1sin)1cos22log2(

2

1

1

1

k

kk

k

.301376553376076650855... 6

2

92

0

1

x xarcsin dx GR 4.523.1

... 12

21

2

2 11

2

1

sin( )

( )!( )k

k

k

k

k k

... 1

5 1 51

0

1k

kk

k

k

( )

... log log( )2

0

13

4

1 2

1 2

x

xdx

... 22

22arctane g d

1 .3018463986037126778... log ( )( )

loge e k

k kk

k k

2

12

111

12

1

= logsinh

log log

1 1i i

... Ik

k

k0 2

0

44

( )( !)

... 840

3 3

2 1

161

k

k

kk

k

... 1

2

1

2

2 1

2 11

logcotcos( )

k

kk

GR 1.442.2

...

0 )33)(13(

)1(

36 k

k

kk

= dx

x

x dx

x x

dx

x30

4 21

2 28 1

( )3

=

012

7

012

3

11 x

dxx

x

dxx

...

2 2

1)(

kk

k kH

2 .302585092994045684018... log 10

...

03

43

)2(3

)(

12

4

32)3(56

k k

=

334

)()2)(1(

kk

kkk

... 2

2

0

1

122

2

1

loglog( / )x

xdx

... log2

3

... 1

2

1

2

1

1e k

k

kk

( )

!

... 2 5 5

2

5

3

5csc sec

... 11

2

1

2

1

2

cos

sinsinh

coscosh

cos

= 1

1

21

21 21

1

cos

sin

( )cos

!(cos ) /e

k

kk

kk

... 4

4 11

1

2

20

2

2 2(log ) log( )

x

x

x dx

x GR 4.298.19

... log!

11

1

kk

... 8

9 3

2 2

3

1

3 2

1

1

log ( )

( /

k

k k k )

...

2 2 2

5

61

2 1

21

1

coth ( )( )

kk

k

k

= 1

2 122 kk

... 21

3

1

2 23 3

2

0

1

Li Lix

x xdx

log

( )( )3

... 2

2 2

0

1

12

14

27 x xarcsin dx

.30396355092701331433... 3

2

... 3 2 21 2

12

1

( )

( )!

k

k kk

!

... 24

2

4

1

16 1221

log

k kk

... 1

2 2

1

22 2log

sinhlog ( ) ( )

i i

= ( )( )

12 1

21

1

k

k

k

k

... log log log ( )2 2 2 22

3

x dx GR 6.441.1

.3053218647257396717... G

3

... 5040 185471

ek

k kk !( )

... 1

23 kk !

.30562962705065479621... 3 2

2

1

31

2

2 1 3 2 11

arcsin( )!!

( )!! ( )

k

k kkk

... 1

3 22 k kk log

... Fk

kk

3

1 6

... 63

2 3 12

2

0

1

( ) log( ) logx x dx

... ( )

( )!

/k

k

e

kk

k

k

2 1

11

1

32

.30613305077076348915... 4 3

27

1

2 12 30

dx

x x( )

1 .30619332493997485204... 3 2

4

3 2

2

3

22

1 4

23 2

3

1

sin cos( )

( )

( )!/

J

k

k

k k

k

13 .30625662804500320717... 128

9

1 3 1 1

42

1

Gk

kk k

kk

( ) ( )( )( )

... 1

4

3

42

0

4

x e dxx

1 .306562964876376527857... 1

24 2 2

8 82

5

8 cos sin sin

= 11

4 20

( )k

k k

... 12

3

1

30

1

21

Jk

k

kk

( )

( !)

8 .3066238629180748526... 69

... 26

405

6

1 11 1 1

4

1 3 41 3 1 3

42 3

( )

( ) ( ) ( )// /Li Li /

= log3

3 21

x

x x xdx

...

122

22

)1(

1

2

1csch

4coth

4 k k

... 1 21

122 2

log( )

( )( )k

k

kk

kk

k

= k

kkk

1

22

J145

= 1

2 11k

k k k( )

J149

= 1

4 22

1

2 21

1

31k k k kk

k

k( )

( )

( )

= dx

x x k k

dx

x xk3 2

12

12

1

1

= dx

e ex x

10

= e ee

x

dx

xx x

x

2

2

0

GR 3.438.3

= ( ) /x e edx

xx x

1 2

0

GR 3.435

= xx

x

dx

x

1

0

1

log log GR 4.283.1

= xe

edx

x

x

20 1

GR 3.452.4

= log x

x xdx

2 21 1

GR 4.241.8

= GR 4.384.5 log(sin ) sin/

x x d0

2

x

x ... GR 4.591.2 2 20

1

log arccos logx x d

... 3 1 2

4

1

4 2 1 2 3

2

0

arcsinh

( )

( )( )

k

kk k k

k

k

= x xarcsinh0

1

dx

.307654580328819552466... 3 3

8 48

1

16

1

2

2 1

31

( ) ( )

( )

k

k k k

.307692307692307692307692= 4

13

.307799372444653646135... 111

1

1

ek e

ek

kk

/ ( )

!

.30781323519300713107...

log(cos )( )

sin1

21 1

2

1

k

k

k

k J523

... arccot

( )

( )

1

2 12 11

k

kk k

... cosh( )

42

16

2

4 4

0

e e

k

k

k ! AS 4.5.63

.308337097266942899991... ( )( )

13 1

1

1

kk

k

k

.308425137534042456839... 2

20

2232

1

4 2

1

2

( )

( ) (

k

k k

kk

k

)

= x x

xdx

x x

x

x

x

log log arctan4

0

1

41

20

1

1 1 1

...

0 03

1

20

2

!

)1(

)!(

)!2()1()2(

k k

k

k

k

kk

k

e

I

...

1

0

1

0

1

0

2

12log6

43 dzdydx

xyz

zyx

...

022

)1(

k

k

k

... 4

... 5

12

...

Li2

1

3

= 1

23 2 3 4 4 2 2

1

42 2

2log log log log

Li

= 1

63 3 3 4 6

3

42 2 2

2

log log Li

= ( )

1

3 4

1

21 1

k

kk

kk

kk

H

k

... 3

2 6

9 3

29

1

3

2

3 21

log

k kk

= ( )( )

12

31

1

kk

k

k

... 24

3 CFG A10

...

1

...

0

2

2

2

)2log(

22

2log3

x

x

... 6e

.309859339101112430184... 1

2 2 6 6

1

6 1

2

621 1

cot

( )

k

k

kk

k

... )()( ii

... log

( ) ( )2

2

1

8

1

2

1

2

i ii i

= sin2

0 1

x

edxx

... ( )( )

( )( ) log

12 1

11 1

111

1

22

2

k

k k

k

k kk

k

... k

Hkkk 21

...

2 2

23

2

3 8

3

8 42 3 coth coth ( )csch csch

= ( ) ( ) ( )

1 22

2

k

k

k k k

... 9

1 116

1

17

1e

k

k

k

kk

k

k

k

( )!

( )!

...

159

1820 2 1414

1

8 221

cot

k kk

... 1

2 2

3

41

32 2

0

2

sin

/

x dx

=

5

43

41 4

0

1

dx

x

... 1

0 k kk ! !!

... 2

11

2 20

e

xe x

dx

log log

... 2

41arcsinh

.311696880108669610301... 22 sinh2

csch16

1

=

122

2

1

1

)14(

4

4

)2()1(

kkk

k

k

kk

.31174142278816155776... sinsin

coshcos

sinhcos

sinsin

(cos ) /

1

2

1

2

1

2

1 1

21 2

e

= ( ) sin

!

1

2

1

1

k

kk

k

k

.311770492301365488... ( )( )

!

11

1

1

k

k

kH

k

... 2

3 3

1 3

2

3

3 3

1 3

22 2

i

iLi

i i

iLi

i

= 5

36

1

6

1

3 1

21

20

1

( ) logx x

x xdx GR 4.233.4

= x

e e edxx x x( )

10

GR 3.418.2

... 7

7

2

7

8

1

5 821

tanh

k kk

... 7

.311993314369948002... 12

3 2

22

2

2

2

log

loglog

=( ) ( )

1 1

11

k

k

k k

k

= 1920

14631

1

71

1

111

1

19

= 2 11

31

1

71

1

111

1

192 2 2

2


... 3 3

4 62 4 2 8

1

2 1

2 1

31

( )log

( )

( )

k

k k k

... 2 1111 2 2 2 2 22 3

1

HypPFQk

kk

k

{ , , , },{ , , , ,},!

= 5

16 5

1

2

1

113 1

23 2

2

k

k k k k kkk k k ( )2

= k kk

( )2 1 11

2 .3126789504275163185... ( )2 1

21

21

2

k

kLi

kk k

... ( )

! ( )

1

1

1

1

k

k k k

... 2 3

13 2

e

x

xdx

cos

( )2

.31284118498645851249... log ( )( )1

22

22

12 1

21

1

i ik

kk

kk

.312941524372491884969... 12

1

6

2

6 41

log arctan x

xdx

... (coth )!

1 12

1

2 22 2

1 0

e e

B

kkk

kk

k

... coth cot( )!

11

1

4

2

2

22

0

e

ei i

B

k

kk

k

AS 4.5.67

... 2

11

22

21

0

e

e

B

kk k

k

k

( )!

... arccsch

... log arctan( )(

34

10

3 2 9 12 21

k

k kk )


... 1

531 kk

... 1

11 1

2 21 1e e k e ke

k

ekk

k

( )

.313261687518222834049... log( )

11 1 1

1

e e

k

kk k

=

1 1xe

dx

1 .313261687518222834049... log( )1 e

...

( )

( )

3 1

2 1

.31330685966024952260... )2(2

1)2(

3

24,3,

2

1F

2

1120211 I

eI

e

=

k

k

kk

k 2

)!2(

1)1(

0

.3133285343288750628...

1

211

)!1(

)!()1(

24

1

32 k

k

k

k

=

2/

04

2tan2

cotcos

cos22

dxxx

xex x GR 3.964.2

.3134363430819095293... 23

42

1

4

1

1 31 1

ek k k kk k!( ) ( )! !

.313513747770728380036... 2 23

23

3

2

1

6 21

log log

k kk

= dxx

x

1

02

6 )1log(

...

1

21

31

3

1

3 31

i i

k kk

=

1

1

3

)12()1(

332

1

kk

k kii

.313666127079253925969... log( )

( )

2

3

... ( )

! (

1

2 2 1

1

1

k

kk k k )

Titchmarsh 14.32.3

... 1

216 3

1

4 1

2

40

1

, ,log

/

x

xdx

16

... 3

3

2

2

3

1

3

hg hg

2 .31381006997532558873... 4

12 372

2 21

2 1

1 1

log( )H k

k k kk

k

...

( )

( )

( )

( )

( ) log2

2

2 2

22 3

2

21

k k

kk

1 .31413511119186663685... 3

4 16

2 4 3

30

x

xdx

sinh

... 10

... 7129

22680 9

1

99

21

H

k kk

... 1036

225 3

22

5 2

H ( )/

... H

k kk

kk

( )

( )

3

1 2 1

... e e Eik

kk

( )( )

( )1 1

1

11

!

... ( ) ( )

1

2

1 3

1

kk

kk

H

... 1

3

1

3

1

3 2 1

1

1

sin( )

( )

k

kk k !

...

0

)2/1(

11

2

3

2 xe

dxxi

... 1

1 51 ( )

kk

... e2

... 3

22

2016 4

24

loglog x

xdx

... ( )( )

( )

13

2

2

2 11

3

3 211

kk

kk

kk k

k

...

1

9 332log

xx

dx

... 3 3

6 3

6 3 3

6

3 3

6 3

6 3 3

65 6

7 6 2 3

5 6

7 6 2 3

i i i i/

/ /

/

/ /

3

3 31

1

35 6 1 3/ /

= 1

3 11

3

321

11 k

kkk

kk

( )( )

... 9

8

15

161

2 51

eerfi

k

k kk

!( )

...

( )log ( )

k

kk

k

2

=6

19

.3158473598363041129... tan x dx

ex0

1

... log( log( ))1 10

1

x dx

... 1

2

1

2

2

0

1

ex e dxx

=

13

152

1cosh

2

11cosh

x

dx

xx

dx

x

... 31 4/

... 10

10

.3162417889176087212... log (loglog

) log (log

)2 32

23

3

2

= ( )( )

121

kk

k

k

k

...

loglog

log( )3

2

1

30

1

1

x dx

x

k

kkk

GR 4.221.3

... 3 3 2 2 ( ) ( )

.31642150902189314370...

1

2411

2 14

)24(

14

)2(

2

1

kk

kk

k

kkk

.3165164694380020839... 1

41

40

2

21

Ik

k kk

( )( !)

... Ei e Ei e dxex

( ) ( ) 10

1

... 11

.316694367640749877787... 2 2 2 2 22 1

81

( log log( )

k

k kkk

=( )

( )!

2 1

2 21

k

k kkk

!!

... 1

3

1

3 3 41

e

dx

x

e

... cosh sinh

( )

1 3 1

16 8

1

16 2 1

4

1

e

e

k

kk !

... arccosh arcsinh2 21

22 3 log

= log3 1

3 1

.3171338092998413134... 1 2 1 2 21

12

1

( ) log log

( ) ( )

( )

k

k

k

k k

... 3

... 12 2

sinh /

... 13

48 24

11

122

1

2 2

50

1

log

log

( )

x

xdx

... e i i3 2 33

2

3

2 / cosh sinh

... 67

120 1

4 3

3 21

3

0

log

( )

x

x xdx

x

e edxx x

.31783724519578224472... 3 2

1 .3179021514544038949... eEik k

k H

kk

k

k

( )! (

11

11 1

)!

=1

11 1 0

1

( )! !loglog log

k kx dx e x dx

k

e

x

... 1

211

1

211

1

2

1

2 12 1 2 1 20

F i i F i ikk

k

, , , , , ,( )

...

2

3

3 3

2 2 3

log GR 8.366.7

... 1

1 11

k k

k kk

( )

( )( )

x

J1061

= x x dsin0

1

... 1

83 22 2 3 2 coth coth csch csch

1

81 11 1 2 2i i i i i i ( ) ( ) ( ) ( )( ) ( ) ( ) ( )1 i

1

= ( ) ( ) ( )

1 2 21 2

1

k

k

k k k

... cosh

( )

5

14

2 1 21

kk

5 .318400000000000000000 = 3324

625 4

2 2

1

F kkk

k

... ( )1 7

6

... ( )1 1

6

... 3

422

2

0

2

ex x cos( tan ) cos

/

dx GR 3.716.5

... ( )

12

1

1

k k

k

Hk

k

.319060885420898132443... ( )( )

( )!cos

12 1

21

11

1 2

k

k k

k

k k

... 4 2 114

3

1

2 2 50

arcsinh

kk k( )

... 41 5/

.319737807484861943514...

log log( )

( )2 2

1

2 12

1

k

kkk

... arccot cot csc ( )sin

1 2 1 12

1

1

kk

k

k

k

... logk

k kk3 2

2

... 2 2

3

1 3

1 3

2

0

1log log( )

x

xdx

... ( )( )

11 1

22

21

k

k k

k

kLi

k k

... 1

11 1

21 ( )!( )

log

!n

k

kk

n

kn

... k

Fkk

1

...

Lik

k

kk

3

1

31

1

3

1

3

( )

... ( )( )

( )!sin

12 1

2 1 2

1 1

21

2 11 2

kk

k k

k

k k

k

... 2

1

4 2

22

1

ie

k

ki

k

logsin //

=i e

e

k

k

i

ik2

1

1

2

21

logsin/

/

...

2

21

4

2

10

cothsin

x

edxx

...

1

2)22()2(k

kk

= 26

81 1 40

2

dx

x( )

.321006354171935371018... e k

k kk

1 4

14 4

/

!

... cot1

2

... 7

48 42

3 2

4

1

1

2 2

21

loglog log( )

( )

x

x xdx

2

... 4

5

2

5 5

2

5

3

5 1 5 20

csc /

dx

x

... 4

3

2

3 32 3

4

31

16

9 1222

2

log ( )

k

k k

kk k

... 2

1

3

1

3 2 1

1

2 10

erfk k

k

kk

( )

! ( )

... 16

63 37

37

2

1

7 321

tan

k kk

.32166162685421011197...

( )

log ( )k

kk

k

1

.3217505543966421934...

1

12

1

)12(3

)1(

42arctan

3

1arctan

kk

k

k

=

1

21

)1(

2arctan)1(

k

k

k [Ramanujan] Berndt Ch. 2, Eq. 7.5

= arctan( )

1

2 1 21 kk


=dx

x21

2

1

... HypPFQk

kk

k

{ , , }, , ,

( )

1111

22

1

4

12

10

... 2 3 4 ( ) ( )

2 .32185317771930238873... 1

2111ijk

kji

... log5

5 4

2

1

F

kkk

k

... log2 5 ... log2 10

... 3

3

2

31

21

1

5 721

tanh

k kk

... F

k

k

k

k

2

1 2

... 2

2212

1

2

1

2

kk

= 2

22

22

24

1

2 Li e Li ei i( ) ( )

= H

k k kk

k ( )(

1 21 )

= ( )( ) ( )

11

21

2

k

k

k k

= ( )cos

1 12

2k k

k

...

12

0 )()(

k k

kk

1 .322875655532295295251... 7

2

... 2 202

0

2

I e x( ) cos dx

... H

k kk

kk

( )

( )

2

1 2 1

... ( ) ( ) ( )( )

2 2 3 41

2

41

k

kk

... ( ) (

!

k k

kk

2

2

)

... 15

8

7

2

1 .32338804773499197945... 9

251

1

3 21

sinh

( )

kk

... ( )

1

3 10

k

kk

... 1

23 2 2 1

11

2

log log ( ) ( )

k

k

k

kk

... sinh( )!

1 1

2 1 2 10

k k

k

AS 4.5.62

... log log( )2

0

15

8

1 4

1 4

x

xdx

3 .3239444327545729800... 121

H kk

k

.323946106931980719981... arccsc

... 1

2 11 2 1

0cosh( ) (

k k

k

e ) J943

.324137740053329817241... 2

2 216 2

1

4

1

2

Li e Li ek

ki i

k

( ) ( )cos

2 GR 1.443.4

... ( )

( )!sinh

2 1

2 1 2

1 1

22 11 2

k

k kkk k

k

...

15

3015

491

770

1

8 121

cot

k kk

... 23 2

0

1

8 21

loglog( ) log

/

x x

xdx

6 .324555320336758664... 40 2 10

1 .324609089252005846663... cosh( )

4

11

4 2 1 20

kk

J1079

... ( )

( )

1

1 3

2

1

kk

k

2 .32478477284047906124...

1 1 11 11

0

1

3

2

1

12

1

12

)(

2

)(

i j kijk

k jjk

kk

kk

kkt

... 3 6

3

1

2

1 12

1

( ) ( )

!

k

kk

k

k

!

... arctan e d

e ex x

2

2 20

1

2 8

x

8 .32552896478319506789...

1

2

12

)(

kk

k

.325735007935279947724... 1

3

... ( )

( )!cosh

2 1

2

11

1 2

k

k kk k

... Hk

k

k

kk 22

1

... ( )

!

k

kk

1

... ( )( )

( )!sin

1 22

2

11 2 1

1

2

1

k k

k k

k

k k

.3264209744985675... H ( )

/2

1 6

... 224

3128

1

16

2 2

0

kk

k

k

... 12

1

2

1

2

1

21

2

21

HypPFQx

xdx, , ,

cos

.32655812671825666123... 1)1(3

)1(62log2

34

1

kk

k

k

1 .32693107195602131840... ( )4 2

2 21

2

41

k k

kk 1k k

... e ( )3

... )(zofzeroofpartimaginary 9 .3273790530888150456... 87

... 17

42 2 6 3

1

2

3 21

( ) ( )log

( )

x

x xdx

.3275602576698990809... 1

32 3 1

21

( log log ) ( )( )

kk

k

k

... 4

3 ( )

... 8 41

1 2

1

1 2

1

2

1

21

Gk k

k k

k

( )

( / )

( )

( / )

... 82

13

41

Gk

kkk

, Adamchik (28)

... 2 56 31

42

13 2

14

30

( )( )

( )

kk

... 1

1 122 k

k

kk

log

... 11

21 1 ( ) ( )i i

= ( ) ( ) ( )

1 22

k

k

k k 1

.32819182748668491245... 1 11

1

22 1 1

1

1 4

2F ( , , )

( )!

k

k

kkk

=( )!!

( )!!( )!

2 1

2 11

k

k kk

... 256

35

4

1 2

/

1 .32848684293666443478... 3 2

2 16

3 3

30

1 log arctan

x

xdx

= 2/

03

3

sin

cos

dxx

xx

... ( )k

kkk 21

13 .32864881447509874105... 23

... 2

22

032 2 2

1

2 2

log( )k

k k

=

0 2/1xx ee

dxx

... 2

30

...

1

0

1

022

2

1

)1log()3(72log2

16

1dydx

yx

xy

... dxxe x )1(2

5

4

3 2

0

2

.32940794999406386902... | ( )| k

kk 41

.3294846162243771650... 1

91 2 2 2 2 3 1

21

( log log ) ( )( )

kk

k

k k

...

3 2

2

1

0

1

loglog

log logx

x xdx GR 4.325.11

... 2

3

39

12

1

1

2 2

0

1

( ) logx x

xdx

... arctanh1 1

2 12 10

kk k( )

... k

k kk 2 2 11 ( )

...

162

23

22

2 22 2 2coth coth

csch csch

= ( )( ) ( )

( )

12

2

2 2 1

2 12

1

2 2

2 31

k

kk

k

kk k k

k

...

2

1

2

1036

225

7

2

( )

... 2 1 11

e Eih k

kk

( )( )

!where h k Hi

i

k

( )

1

... 45 5

2 1

4

0

4

21

( ) log

x dx

e

x

x xdxx

... log( )e 2

... )1sinh1cosh1cosh()1sinh()1cosh(2

1 ee

)1sinh1cosh1sinh(2

1

= e e

e

k

k

e e

k

2 1

12 1

/ sinh

( )!

... 1 4/

...

32

3

1 2 3

3 44

,( )( )( ) (

k k k kk

k

)

... 1

21 1 1 1

0

1

cosh sin cos sinh sin sinh x x dx

.33194832233889384697... 4 4 3 1 32 2

0

dx

x x( )( )

...

2 2

3 3

2

3

2

1

6log

log Berndt 8.6.1

... ( )3 1

2 213

1

k k

kk 1k k

.33233509704478425512... 3

16

1 3

1

1 12

1

( ) ( )!

( )!

k k

k

k

k

.33235580126764226213... ( )4 3

4 14 31

kk

k

... 4 2 .33302465198892947972... log 3 2 ... 2

.3333333333333333333333 = 1

3

= 1

3 3

1

3 1 3 4

1

2 1 2 51 00k k k k k kk kk( ) ( )( ) ( )(

)

= 1

5 6

1

4 421

21k k k kk k

3

= ( )

( )

1

21

22

1

1

1

1

k

kk

kk

k

kk

k

=

11 19

3)2(

13

)(

kk

k

kk

kk

= 2

2 5

120 1

k

k kk k k

kk

k

!( )( )

= k k

k kk

( )

( )( )

3

1 21

J1061

= 12

11

4

2 122

1

k k kk k( ) ( )

J1061, Marsden p. 358

= dx

x

x

xdx

( ) ( )1 140

2

40

=

0

5

0

2

33 xx e

dxx

e

dxx

= sinh x

edxx2

0

1 .333333333333333333333 =

0

2

)2(4

1

2

1,2

3

4

kk k

k

k

=

122

11

kk

.333550707609087859998... log csclog ( )

2

3 2

2

2 1

21

k

kkk

.33357849190965680633... 1

4 11k

k

( )

.33373020883590495023... sinh sin1 1

...

6

12 2

( )

.33391221206618026333...

0

1

12

2log

8

1

312812

)1(

k

k

k

... ( )k

kk

1

2 11

... 3 2

4 16

2 2

21

log arctan

Gx

xdx

...

10

51

50

1

2520

coth

kk

.33432423314852208404...

84 4 2 4 1 6 2 arcsinh log \

= arctan arcsinx x0

1

dx

... 622

2

1

ek

k

k

k

!

... log( )k

kk

22

1

... e /2

... 2 2

... 11

21

kk !

... 41

41

2

1

40

1

7 221

tan

k kk

... e

ee e e

k

e kkk

11

10

log( )( )

... 6

720 volume of unit sphere in R12

...

( )2

21

kk

k

... 3

3

... log log ( )( )

2 2 12

11

2

kk

k

k

k

= 1

2 11

21 k kk

log

... 2

2e

= 42

125 6

2

1

kk

k

... 46

75

2 2

5

1

2 51

log

( )k kk

... 4 41

1 2

1

21

Gk

k

k

( )

( / )

...

22

2

)()1(

log

kk

kkk

k

.336409322543576932... ( )k

k kLi

kkk k

1

2

1 1

221 1

2

... Li12

7

... 2 3

3

2

120

12

3

2

32

1

2

log log log

( )

Lix

xdx

... e

e2 1

.33689914015761215217... 14

3 32 2

1

2

2

0

2

log( cos )

sin

/ x

xdx

... ( )

( )!/2 1

11

1

2 1 2

2

2k

kk e k

k

k

k

... Likk

k5 5

1

1

3

1

3

... 3

2 4 2 3 33 1

12

121

log log( )!

( )!( )

k

k kk

... ( )

( )!!

/2

2 21

1 2

21

2

k

k

e

kk

k

k

...

1 21)2(

k

k

kk

... 4

8 6 2 CFG D1

... k

k kk

1

2

... 6

25 5

1

2 1

1 12

12

1

( ) ( )!( )!

( )!

k

k

k k k

k

=( ) ( )!( )!

( )!

1

2 2 1

1 12

12

1

k

k

k k

k

...

2 2

3

3

21

3

2 22

sinkk

... )1(Re)2()!2(

)1()1(

1

Eikk

cik

k

AS 5.2.16

1 .337588512033449268883... 2

0

1

48 4

2

2

loglogarccot x x dx

... 2 11

2 11

log/

e

dx

ex 2

.33787706640934548356...

3

22

2 1

11

log( )

( )

k

k kk

... ( )

( )( ) log

2

11 1 1

1

1

22

22

k

k kk

k

kk k

= 2log2

1

... 2

3 21

3 216

2 21

logk k

dx

x xk

... ppf k

kk

( )

22

.33868264216941619188... 68 1

51

ee

x

dx

xcosh

... 1

1 20( ) !e

B k

kk

k

... 18

241

12

0

1

x

xx dxlog log

=log( ) logx x

xdx

13

1

... 12

2 3 14 3

1

22

2

2

32

( ) ( )

( )k k

k k

k kk k

1

...

2 2

8

3

2 1

2

0

2

x dx

ex

.339395839527289266398... ( )

log2 1

4

11

1

412

1

k

k jkk j

j

... 16

15

2

1 2

/

3... 761

2240

1

1625

kk

... e

8

... 2

31 2

1 3

21 2

1 3

2

2 31 3

1 32 3

1 3

//

//

/( ) ( )

i i

2

32 2

2 32 3

//

= 4 3 14

4321

k

kk

kk

( ) )

.339836909454121937096... arccsc 3

... ( )

!( )!

k

k k kI

k kk k

1

1

12

1 1

21

2

... 2

1

( )

!

k

kk

... 2

29

... 1

9

2

3

4

271

1

9

4

3

4 2

9 2

1

3 11

21 2

21

( ) ( ) ( )

( )

g

kk

= log x

xdx3

1 1

... ( )

!( )

k

k kk

1

12

.340791130856250752478... Likk

k4 4

1

1

3

1

3

.34084505690810466423...

1

2

1 1

1

1 12

1

( ) ( )! ( )

( )!

k k

k

k

k

1 .340916071192457653485... 1

21 F kkk

1 .340999646796787694775... 1

3

5 3

27

12

11

k

kk

.34110890264882949616... 27

1 2

1

1 12

1

( ) ( )!

( )!

k k

k

k

k

... 9

16

1

2

4

0e

k

k

k

kk

( )

!

... ( ) logk kk

1 11

... 5

2

15 2

1

kk

/

.34153172745006491290... 1

2

1 3

26 31

1

1k k

k 3

k

k

kk

( ) ( )

4 .341607527349605956178... 6

... 3

5

.34176364520545358529... 1

2

1 4

26 21

1

1k k

k 2

k

k

kk

( ) ( )

... 2

2

1122

2 1

log( )

H

k kk

kk

80 .34214769275067096371... 43 13

0

ek

k

k

k

( )

! GR 1.212

... log ( ) k

kk

2

2 .34253190155708315506... e e

... k k k

kkk k

( )

( )

3 1

2

2

2 11

4

3 21

.3426927443728117484... ( )4 1

4

1

415

1

k

k kkk k

=

1

41

1

21

21

1

21

2

i i

52 .34277778455352018115... 945

32

11

2

.343120541319968189938... 8 2 3 41

4 5 31

log ( )

k kk

2 .343145750507619804793... 8 4 2

... 3152

2796 3

1

2

1 40

1

( )

log/

x

xdx

... sin

( )

x

xdx

1 20

.3433876819728560451...

0

1

12

2log

4

1

312412

)1(

k

k

k

... H

k kk

k

2

1 2 2( )!( )

... 12 6 1 10 11

2 20

cos sin( )

( )!( )

k

k k k

.3436037994206676409... 1 1

10

e

e

e

k

e k k

k

( )

( )!

... 1 1

10

e

e

e

k

e k

k

( )

( )

k

!

1 .3436734331817690185... 1

2 311

kk

... 2 4

2

7

12012 2

9 3

2 log

( )

=

log log1

1

0

13

xx dx

16 .344223744208892043616... k k

kk

3

2

( )

!

... 2

3 2118

11

54

1

3

k kk

... H k

kk

( )3

1 4

2 .34468400875238710966... 1

4

1

2 10

e erfie e

erfi ek

k kk

cosh

!( )

.344684541646987370476... 265 72061

ek

k kk

!( )

.344740160430584717432... 2

12

2

20

2

cos

cos

x

xdx

5 .3447966605779755671...

csc5

1

5

4

5

= log 1 5

0

x dx

...

4 1 40

2

ex e x dxx

/ sin

.3451605044614433513... 9 4

11900001

2564

5

sinh

kk

.34544332266900905601... sin cos

( )( )

1 1

41

2

2

1 !

k

k

k

k

... 1

2 22 1 2

0

e ek

ke e

kk

/ / sinh

!

... ( ( ) ) log k kk

1 2

2

...

1

2 2 2 2

2 1

2

1

2 112

2

cot

( )k

kkk k

1 .345506031655163187271... 1

2 2 2 2

2

2

1

2 112

1

cot

( )k

kkk k

...

logcos

sin

arcsin/

2 21 10

2

0

1

G

x x

xdx

x

xdx

= K xdx

x( )

20

1

GR 6.142

.345729840840857670674... 1

32

1k

k

.3457458387231644802... 1

2

2

2

1

1

2 10

20

I

e k

k

k

k

k

( ) ( )

( )!

... 1

21

21

21

1eEi

H

kk k

kk

log ( )!

... 2

1

3

1

3 2 12 10

erfik kk

k

! ( )

... 2

0

1 x

xdx

... ( ) sin ( ) ( ) sin ( )/ / /1 1 1 13 4 1 4 1 4 3 4 /

... 5

5

2

1

5

1

5 521

tan

k kk

.34649473470180221335...

( )

( )

( ) log2

22 22

k k

kk

.34654697521433430672... 4

32

5

21

11

11

20F

k

k

k

k

, ,( )

( )

!

... logRe log( ) Re ( )

2

21 1 i Li i

= arctanh1

3

1

3 2 12 10

kk k( )

AS 4.5.64, J941

=

111 2

1

2

1

2 kk

kk

kE

LiH

= ( ) cos /

1

2 2

2

0 1

k

k kk

k

k

=

( )( ) log

2 1

21

1

21

1

1 12

2

k

kk

kk k k

= dx

x x

dx

x x( )( )

1 313

1

= x

x x xdx

1

1 120

1

( )( ) log GR 4.267.4

=

002

112xx

e

dxx

e

dx

=

4/

0

4/

0 sincos

sincostan

dxxx

xxdxx

= ( )cos

10

ex

xdxx

= GR 6.496.2 ( ) sin sinx x x30

1

dx

= x x

xdx

2

20

sinh

cosh

GR 3.527.14

.34700906595089537284... ( )

11

13

1

k

k

k

k

... 2 2 2 2 22 2

2

1

log log( )

H

kk

kk

... Likk

k

1 2

1

1

2 2/

.347296355333860697703... 218

2 2 2 2sin ...


... 6 5

04 1 1

log

( )( )

x

x xdx GR 4.264.3

.347312547755034762174... 24 5 12 32

5 1

4 4

30

( ) ( )( )

x

ex

.347376444584916568018... 32 2131

6

1

2 50

log( )

kk k

.347403595868883758064... erf1

... ( )

logk

kk

k

1

2

.34760282551739384823... 4

3 3 2

1

2 2

2

20

2

cos

( sin )

/ x

xdx

...

2

32 2 1 2

1

2

3

2

1 31 3 1

2 3 2 3

// / /3

/ /( ) i

2

31 2

1

2

3

2

1 32

2 3 2 3

// /3

/ /( ) i

= ( ) ( )

1 2 3 12

21

13

2

k k

k k

kk

.3477110463168825593...

1

1)1()(k

kk

1 .348098986099992181542... 3

23

... 2

3 902 3 16 2 16

1 4

2

2 4 1

1

( ) log( ) ( )k

kk

k

= 1

2 5 41 k kk

1 .348383106634907167508... i i log

7 .3484692283495342946... 54 3 6

7 .348585884767866802762... 1

33

1

2

3

21

cot tank kk

Berndt ch. 31

... Likk

k3 3

1

1

3

1

3

... 9 1

7 2

90

x dx

x

/

.3497621315252674525... 42

3 21

4

1 1

421

1

1

log

( ) ( )

k k

k

k

k

kk

= hg1

4

5

4

= 1

0

4 )1log( dxx

... primepp 12

1

... coth21

21 1 2 1 2 i i

= 4 1

41 4 2 2 13

1

1

1

( )( ) ( ) ( )

k

k kk k

k

k

k

k

... 1

2

5

3

1

3

2

31

1

3

2

30 1

3

0

1

, , , e dxx

... k

kk

k2

2 11

( )

= 20

7

... 11

220

k

k

...

4

1

2

1

2 1

2

20

4

arctancos

cos

/ x

xdx

.35024515950787342130... | ( )| 2

4 11

kk

k

.35024690611433312967...

6 3

2

3

3

6 131

2

log log x dx

x

... 2 1 2 21

2 11

sinh( )

kk !

... ee kk

12 1 2

1

2 11

sinh( )!

dx

= GR 3.915.1e xxcos sin0

... E1

2

... 52 42 15

1

2

1

ek

k

k

kk k

!

( )

! Berndt 2.9.7

.350836042055995861835...

2 1 31 3 2 3

2 3 2 3

6

2

61 2 1

4 2 2 3

4

/

/ // /

( )i

( ) /

/ / /

12

6

4 2 2 3

41 3

1 3 2 3 2 3

i

= ( )( )

13 2

2

1

21

5 211

kk

kk

k

k k

1 .3510284515797971427...

( )( )

( )

3

2

3

42 1 1

4 3

12

2

4 2

2 32

k kk k

k kk k

1

.351176889972281431035... 2 2

24

2

8

log Berndt 9.8

15 .351214888072621297967... ( ) ( ) ( ) ( ) ( )2 6 4 6 3 1 13

2

k kk

= k k

kk

2

42

4 1

1

( )

.351240736552036319658...

4 5 4 520

dx

x

.351278729299871853151... 21 20

ek

k kk ( )!

... K Kdk

k2

0

12

2

( )k GR 6.143

.35130998899467984127... 1

2

1 4

251

1

1k k

k 1

k

k

kk

( ) ( )

... dxx

x

1

0

22

1

arcsin

2

3216

... 4 2 3 1 3 121

e e eEi EikH

kk

k

( ) ( )( )!

... ( )

logk

k kkkk

1 1

2

11

1

221 k

... e2

.3520561937363814016... 1

4 1 422

11 ( )

( )k k

kk k

k

... 2 31

4 2 1 2 30

arctanh1

2 kk k k( )( )

... 1

3

1

3

1

3 2 11

sinh( )

kk k !

... 22

11 2e isiin ei ei i

( ) / cos cos ( )i

... 4

4

... 1

33

12

13SinhIntegral

x

dx

x( ) sinh

... 2

28

... arccot e ee k

k

kk

2

1

2 12 10

arctan( )

( )

= dx

e ex x

1

... 1

22 2 1

2

12

20

e Eik

k k

k

k

( ) log!( )

...

1

02)1(

)2()1(2

11

xe

dxEiEie

e x

... 54 193

3

1

ek

k kk !( )

... Jk k

k

k

k

k

k

k2

0

3

20

21

21( )

( )

!( )!( )

( !)

LY 6.117

2 .35289835619154339918... ( )k

k kLi

kk k

1 12

12

1

1

1 .352904042138922739395...

22

3172

5 4

4

1

25 4 2

( )( ) ( )

H

kk

k

Berndt 9.9.5

.3529411764705882 = 6

17

.35323671854995984544... 1 1 11

2

p k

kp prime

probability that two large integer matrices have relatively prime determinants Vardi 174

.35326483790071991429... 2

1 100

1

( ( ) cos ) sin arcsinJ x x dx

.3533316443238137931... ( )

1

2 1

1 2

1

k

kk

k

... x dxx1

0

1

/ 1 .353540950076501665721... ( ) ( )3 2

.353546048172969039851... cos sin sin1 1

2 1e e

x x

edxx

... 8

sin8

3sin

8

8

4

2

...

4

1

2 6 2

12

4 21

4 21

cothk k

dx

x xk

...

1 5

)2(

5csc

5log

kk k

k

... log( ) ( )!( )

e Lie

Lie

B

k kk

k

1 1 21

21

2 322 3

0

= x dx

ex

2

0

1

1

... 2

3

.35417288226859797042... ( )4

4

1

4 114

1

k

kkk k

= 1

2 4 2 2 2

cot coth

... ( )( )

!

12

k

k

k

k k

.354248688935409409498... 3 7

.35429350648514965503... 1

431 kk

.35496472955084993189... 11 1

2

2 1

40

1

1

e

Ik

k k

k

kk

( )

!

... 2

2 12

1 2 0

1

log arctan arccosx x dx

.355065933151773563528... 2 21

1

4 1

2 2 121

2 21

( )( ) ( )k k

k

k kk k

=

212

23)1()()1(1)2(

1

k

k

kk

kkkkk

= ( ) ( )( ) ( )

1 21 1

21

1 2

k

k

k

kk

k kk k

= log GR 4.221.1 log( )x x10

1

dx

.355137311061467219091... 7

1920

4 3

31

log x

x xdx

4 .355172180607204261001... 2 2 log

= x dx

ex

10

GR 3.452.1

=

02

2

1

)9log(dx

x

x

= x x d

x

sin

cos10 x

GR 3.791.10

=

2

0 2sinlog dx

x

= GR 4.224.12 log ( cos )2

0

1 x dx

= GR 4.226.1 log (cos )/

2 2

0

2

x dx

... 5 3 1

6 2 1 12 50

dx

x /

.355270114150599825857... 3

2

2 1

11

11

1

22

2

log( )

log k

kk

kk k

.35527011687405802983... ( ) ( ) tanh3 2

3

3

2

15 4

1

3

k k kk

.3553358665526914273...

1

2

)!2(

)2()!(

k k

kk

.355379722375070811280... 1

3

2

3

3

2

1 4

32 2

1 30

2 3 1 3

e ek

k k

k

/ /

cos( )

( )!/

... )3(

)6()3(

.35577115871451113665... 1

1342 kk

1 .35590967386347938035... 11

41

kk

... coth kk

11

... 3

41

22

1

arccot ( ) arctankk

K Ex. 102


=dx

x x20 2 2

... log7 2

... 4 2

1

21

2

4

1

4 11

211

coth ( )( )

kk

kk

k

k

.356395048612456272216... 1

42 2

3

2

1

8 22

2 11

Fkk

kkk

, , ,

... ( )k

kk 41

.3565870639063299275...

( )

( ) ( )

( )12

23

16 1

11 2

3

k

k

J1028

.356870185443643475892... 4

3

1

4 4

1

3

1

4 12

2

12

0

LiH

kk

kk

kk

( )

( )

= 5

14

...

0 12

1

3

123xe

dxx

1 .3574072890240502108... kk

k

2

1 3 1

... 3 3

224 2

27 3

2

log

log

=1

1 12

130 ( )( )( )k k kk

.357907384065669296994... 1 11

2

1

21

cot tank kk

Berndt ch. 31

= csc1

22

0k

k

k

Berndt ch. 31

... 36195

2 3

4

0

ek

kk

( )!

... 2

21

1

2 21

sin

kk

1 .358212161001078455012...

0

2 )tan2sin(1sinh

12 x

dxx

ee

=

0

2/

0

cot)tan2sin(cos)tan2sin(

dxxxx

dxxx

1439 .358491362377469674739... 61

128

7 6

0

x

xdx

cosh GR 3.523.9

4 .358830714150528961342... 3 32

( )k

k

... 2

1

2

8

( )kk

k

... 19

...

13 14k k

k

...

00 xe

dxx

xe

dxxx

.359140914229522617680... e k

k kk21

2 41

!( )

1 .359140914229522617680... e k

k

k k

kk k2 2 1

1

21 1

( )!

( )

!

= dxex x

0

1 2

=

1

1 30 e x

dxx ( )

= log

( log )

x

xdx

e

1 21 GR 4.212.7

.359201008345368281651... 3 3 3 4 ( ) ( ) ... 3 4/

3 .35988566624317755317...

1

1

k kF

... ( )

!/k

ke

kk

kk

11

11

22

... arccot ( cos )cscsin

2 2 22

21

k

kkk

=

1

31

2)1(

25

34

kk

kk kF

.360164879994378648165...

1

16

1

2

3

2

1

16

1

2

3

2

i i i i i

i

= ( )( )

14 2

4 41

2

41

kk 1k k

k k

k

1 .360349523175663387946...

12 9/1)12(

1

4

3

k k

GR 1.421

... 11

81

k kk ( )

.360553929977493959557... 1 2 3 4 ( ) ( ) ( )

...

0 )2(2!

)1(64

kk

k

kke

...

12 64

1

4

32coth

22 k kk

... 1

4 2 22

1

2 321

coth

k kk

=

12 45

1

36

13

k kk

= 3)2(

36

49H

...

e Eidx

e xx2

0

22

( )( )

... 2

1arcsin

...

1

32 84

1

1621

coth

kk

J124

.361594731322498428488...

1

2

)1(32

3log

2

3log

2

3

kk

k

k

kH

5 .3616211867937175442...

3 2 2 2 2 3

2 4

6 2

4

3 2

4

2

23

( ) log log log( )

=

log3

20

xdx

e x

441 .361656210365328128367... 162 11

7

1

ek

kk

( )!

...

0

3

)!1(

3

3

1

k

k

k

e

... 1

12 2 2 1 2

42 3 2 3 1 3 2 3

3

32

( ) ( )/ / / /

k

kk

= )4)(1(4

12

105

38

0

kkk

kk

k

.3620074223642897908...

1 )12(

sin

k k

k

... 275

6 3

1

1 3

3 1

31

( )

( /

k

k k )

... ( )( )

( )

12

2

1

4

112

12 2

1

k

k k

k

kLi

k

... Ik k

k0 2

1

2

31

1

3

( !)

...

1 )12(2

11

kk

...

11

1

)(

))((

)(

)1(

kk

k

kp

kp

kp

, p(k) = product of the first k primes

.362360655593222140672... 1

22k

k k( )

... 1

31 ( )k kk

.3625370065384367141... 29

15 21

4

2 1

1612 1

1

( )

( ) ( )kk

kk

k

k k

... erfk k

k

kk

1

3

2 1

3 2 12 10

( )

! ( )

... 35

.3632479734471902766... 1

23 3

2

22 2

0

1

(log loglog( )

x

xdx GR 4.791.6

... 4

3

3

2

.363297732806006305...

sin( ) sinh( )

3 3

241

92 4

2 k

...

1

01 log

1log

)1log(

x

dx

xe

e

ke

k

kk

GR 4.221.3

... 12 2 1

2

1

2 1

2

1

( )!!

( )!!

k

k kk

J385

= ( )( )!!

( )!!( )

12 1

24 11

3

1

k

k

k

kk

.36363636363636363636 = 4

11

... 1

12 k kk ! !

... 1

21 Fkk

... 2

23

1

7 3

41 1

2

4

2 1

( )( ) ( )

( )

( )k

kk k

k kk k

k

...

1

33/1

3!1(9

173

kkk

ke

...

234

234

1

2

32log)2()3(

xx

dx

kkk

.363985472508933418525...

01

2

1

1

sin

1

)1(

sinh22

1x

k

k

e

x

k

sin

cosh

2

20

x

xdx

...

12 26

17cot

7214

1

k kk

...

1 )!1(

11

k k

...

0 )12(!

22

22

1

k

k

kkerfi

...

1 )!2(

42log)2(2

k

k

kkalCoshIntegr

.36481857726926091499... 12

2 21 21

1

log( ) ( )

k

k

k

.36491612259500035018... 3

3 6 7 CFG A22

... ee xx

4

2 2 1

0

2

log log dx

...

0

2 4/1

)1(

2csch2

k

k

k

.3651990418090313606... 31

3 1

1

3

1

332

1

1

( )( )

kk

=( ) ( )

1 1

3

1

1

k

kk

k k

1 .365272118625441551877...

1 1

k

k

2 .365272118625441551877... 1

1

.365540903744050319216...

2

31

2

31

3

1

27 22

2 iLi

iLi

64

9

8

27

64

272

31 2

22

0

log ( )/x Li x dx 26 .36557382045216025321...

.365746230813041821667...

22

11log

11

1

)1(

1)(

kk kkkkk

k

...

22 22

11log

11

11

)(

)1(

)(

kk jj

k kkkk

k

kk

k

22 1

2log

k kk

k

...

1 .365958772478076070095...

11

1 11

2)12()!1(

)2()1(

kk

k

kerf

kkk

k

13

3

...

= cos sin 6 6

1 12

= 6 31

( )k

k k J1029

.36602661434399298363...

( )3 1

2 13 11

kk

k

1

4

1 1

1 1

3

2 962 1

3 4

3 4 23 4

0

4

log( )

( )( ) sec

/

//

/

G i

iLi x x dx

...

log 3

3 41

H Ok

kk

...

036/13

3/1

63

2

x

dx ...

3

23log3log2log)2(

3

12

22 LiLi .36621322 99770634876...

= Lik

H

kkk

kk

kk

2 21

1

1

1

3

1

3

1

2

( )

.36633734902506315584... 3 22 4

7

9

8 1

4 1

2

22

log( )

Gk

k kk

= ( ) ( )k k

kk

1 1

41

1

log log 2

4449934445032...

...

1

1296

1

3

5

6 13 3

3

31

( ) ( ) log

x dx

x .36656347

2 1 1 1 2 1371

97202 3

42 3

41 3

42 3

4

( ) ( ) ( ) ( )/ / / / Li Li Li

=

70 8 .3666002653407554798...

... 1160703963

8388608 9

9

1

kk

k

( ) ( )3 2 4 ...

22

0

1

16

1

4 x xarcsin dx ...

00 cosh12

)1(2

4

3,

2

1

4

1,

2

1

xx

dx

k

k

...

4

G ...

134

1

12

2

1

2

)3()1()3(4log4

38

kkk

k

kk

k ...

.367042715537311626967... cos ( )( ) ( )

2

31

32

323

2

1

4

k

kLi e Li e

k

i i

.367156981956213712584... ( )( )

12 1

1

1

kk

k

k

k

k ks d

k

1

132 2

3

lo ... s( )

g

1 .367630801985022350791... ( )k

kk 21

1 1

22

0 0ei

k ki

k

k

k

k

/ ( )

!

( )

( ) ...

1

!

= ( )

!

( )

!

1 13

0

4

0

k

k

k

k

k

k

k

k

1

2 1 2 30 ( )!( )k kk

=

= ( )k

ekk

11

= k k e

k e kk

( )

( )(

1 11 )

J1061

dxex

x

e

dxx

02

1 )(

log =

= 3

1

1

0

1sinhsinh

x

dx

xdxxx

( ) log ( )

11

k

k

k ...

sinh

( )! (

2

2 2 2

2

2 1

2

2

2 2

0 1

)!

1 .36829887 20085906790...

e e

k

k

k

k k

GR1.411.2 k k

=

122

21

k k

... 2k

2

)1( k

kk

= 7

19

( )

( )

( )2

3 31

k

kk

... Titchmarsh 1.2.12

...

)1()1(2

1)(Re )2()2()2( iii

=

0

2

133 1

cos

)(

1

)(

1dx

e

xx

ikik xk

.368608550929364974778... 2 2

2116

2

2

2

16

7 3

16 4 2

log log ( )

( )

H

kk

k

1 3!

1

k

k

kk

...

2

1 100

1

( ( )) sin arccos J x x dx .3688266114776901201...

02 3/1

1

3coth

2

3

2

3

k k

...

.36966929 9246093688523... 12

2

21

1

21

1

log( )

( )k

k

k

k

= 1

21

11

1 kk

kk

log

3 3

4

33

2

31

4

9 622

2

log ( )k

k k

kk k

...

0 2 ),2(

!

k kkS

k ...

6 2 1 1

2

120

8

120

x dx

x

x dx

x ...

363

980 7

1

77

21

H

k kk

...

( )

( )

3

3 1

1

313 3

1

kk

k kk .3704211756339267985...

dxx

x

1

03

6 )1log(

344

3log3 .3705093754425278059...

2 .370579559337007635161...

2/

0

224

cot)tan2(sin1

34

dxxx

e GR 3.716.11

2

1

k

kk k

...

5

21 1

1

4 120

log( )x

dx .3708147119305699581...

1 )!2(

33sinh

2

3

k

k

k

k ...

4 1

220

1Gx x

sec ... dx

55290132347...

1

22

2

1)2(2

2csc

82

2sin1

kk

kk .37133401

=

222

2

)12(

2

k k

k

2 .3713340155290132347... 12

2 8 2

2

2

22

1

sincsc

( ) k kk

k

= 2

2 1

2

2 21

k

kk ( )

.37156907 160131848243... G( H

k

kk

k

log ) /2

4

1

2 1

11 2

0

Adamchik (25)

=

12 )12(4

)2()14(

kk

k

k

k

= dxx

x

1

021

)2log( GR 4.295.13

= dxx

x

1

02

2

1

)1log( GR 4.295.13

=

0 cosh

)2/cosh(logdx

x

x GR 4.375.1

.3716423345722667425...

1 2 215 2

6

2 2

3

3

2

2 3

( ) log

log log ( )

= log ( )

( )

2

2 20

1 1

1

x

x x

1

44 2 1 4 1 22 (cos ) log( sin ) ( ) sin ... 2

=cos

( )

2

1 1

k

k kk

1

1

2)1(

2arccsch55

4

5

1

k

k

k

k ...

87240681867... 1

4 121 k kk !( )

.37218495

( ) sin logcsc log sin 2 1 1 1 22 ... 2

=

sin( )

( )

2 2

11

k

k kk

GR 1.444.1

1

2

1 3

241

1

1k k

k .37252473265828081261...

1

k

k

kk

( ) ( )

0 )1(8

12221624

kk kk

k ...

... 1

41 12 2 1 1 coth ( ) ( )( ) ( ) csch i i i i

= ( ) ( ) ( )( )

1

12 2k

( )

1 2 211

1 2

k

k k

k k kk k

...

0

22

3

12log2

3 xe

dxx GR 3.452.2

.3731854421538599476... 1

4 1 41

11 ( )

( )k k

kk

k

k

...

.37350544572552464986... 1

2 1

1 4

241

1

1k

k

k

k

kk

( ) ( )

16

1

46

1

3

2log

33 1

kk J80

k

...

= ( )

1

3 20

k

k k K ex. 113

= dx

x

x dx

x

dx

e ex x31

30

1

201 1

.37382974072903473044... 20

9

8 2

3

2 1

4 2 20

log

( )

k

k kkk

( ) ( )log3

3

31

3k

kk

...

2/

0

22

cos12log

44

dxx

xG ... GR 3.791.5

...

4 2 2

61

122

0

1

loglogx

xdx

14

2 )1log(dx

x

x =

( )

( )

1

2 2 1

1

1

k

k kk

k ...

0 )4(2!

15896

kk kk

e ...

02

)1(

)(

1)(

k k ...

1

1

1 1

23 20

2 2 3 122k k k k k k k

1

k kk

1 .37427001649629550601...

ii i i i

1

41( ) ( ) ( ) =

1

2 4 2

1

4

coth ( ) ( )i i =

( )k

kk

1 1

121

...

12 e

...

cos log1 2 ...

1

22k

k k ( )

...

0

/1

!

11

kkk

e

...

/1e ...

...

1227

222 xx

dx

xx

dx

... ii ee 222

11

1)1()cos()1(1

1

kkk

k =

1 2 3 13

01

21

Jk

kk

k

( ) ( ... )( !)

00000000000003

82 1 3

0

e k

k

( ) log .37500000 =

= )4

1 2(

1

k kk

=

13

3log

x

dxx

2 2 2

404 24

2 2

22

log

loglog xdx

e x 1 .3750764222837193865...

dxx

220

1

.3751136755745319894...

ii eEieEi2

1 ...

1 !

cos,0,0

2

1

k

ii

kk

kee =

4

105

1

1

2 2

2 4

p

p pp prime

...

1

1

1

2 21k kkk

j kkj

...

Likk

k

1 2

1

1

4 4/ ...

4

2log2

G ...

5cot

...

2 1 2 2 21 2 1

2

1

1

log log( ) ( )!!

( )!

k

k

k

k k .37645281 29191954316...

1

1 2

4

)1(

kk

k

k

k

k =

12 54

1

5

6coth

2 k kk

...

0 )3()!12(

148

659

k kkee ...

11

2 2 2 2 11

arctanh1 k

kkk ( )

...

=( ) ( )!!

( )!!

1 2

2 1

1

1

k

k

k

k

1

22 2

0

1

G x x xlog ( ) sin GR 6.468 d 1 .37692133029328224931...

13

2

)2(

1

16

3

242

)3(

k kk

...

1

1

1

2

1

2

1

41 1 2ek k ( ) /

1e k

tanh ...

1

2

1

2 1

1

221

( )!k

B kk

= J134

1 .37767936423331146049... 2

33 4 1 2

22

2

( ) ( ) ( ) ( )k

k

k k k

= 4 2

1

3 2

3 22

k k k

k kk

( )

1

...

22

log1

k k

k

7

7 ...

ex x

e(cos sin )log coslog

1 1 3

23

1

... dx

... e

x dxe(cos sin )coslog

1 1 1

2 1

.378139567567342472088... 2 42

3

1

2 20

log( )

( )

k

kk k

12 35

1

2

13tanh

131

k kk

...

dxx

xHypPFQ

13

sin

44

1,

2

3,

2

1,

2

1 ...

dxx

xsici

02)1(

cos1cos)1(1sin)1(

2

1cos1

...

.378605489832130816595...

...

( ) ( ) ( )

1 11

2

k

k

k k

...2

1

2

1

2

1

1 212

12

kkkk kk

0

/1

!

cosh)1(

2

1

k

kee

k

kee ...

2 23

22

2 1

2 1 22 2

1

log log

H kk

k

...

3

2 32

1

5 30

log

( )

/

k

k k ...

3 3

22 6 2

42

23 23 2

2 2

( )log log log 4

x

...

= a rcsin logx x d3

0

1

2

26 ...

( ) ( )

( !),

12

11

12

20

1

k

k k

k

kJ

k k

...

4/

0

6

sin1

cos

20

1

40

2

32

3 dx

x

x ...

log( ) ( ) ( )

!

2

1

1 1 2 1

1

e

e

k

k

k k

j

...

dx

ex

10

=

.37989752371128314827... 3

3

0

1

32

3

16 x xarcs dx in

... ( ) ( )

1 12

2

k

k

k

( ) ( )

1 12

2

k

k

k ...

.38079707797788244406... e

e

B

k

k kk

k

2

2

2 1 22

1

1

2 1

2 2 1

2

( )

( )

( )!

88 2 22 9 .3808315196468591091...

2

53

10

55

2

53

10

55

2

5tan

51 ...

=

2

22 )1(

11)()1(

kkk

k

kkk

kkF

0 1

sindx

e

xxx

...

1 .381713265354352115152... 1

21 ( !!)kk

cos sin ( )( )

( )!1 1 1

2

2

2

1

k

k

k

k ...

log ( )( )

12

12

12

41

1

i i

kk

kk

k ...

= log 11

4 21

kk

.381966011250105151795... 23 5

2

1 2

121

( ) ( )!

( !) ( )

k

k

k

k k

2 .381966011250105151795...

0 2

1134

k kF GKP p. 302

.382232359785690548677...

1 54

5log1

16

5

kk

kkH

( )

1

2 22

k

kk

...

( )

!/3 1

11

1

1

3k

kk e

k

k

k

...

12

1 15

1

5

)2(

5cot

522

1

kkk k

k ...

.38259785823210634567... dxx

x

14

arcsinh

6

)21log(2

( )3

...

.38268343236508977172... 8

3cos

8sin

4

22

.382843018043786287416... ( )( )

14 1

1

1

kk

k

k

... log

( ) ( )2

2

1

8

1 2

2

1 2

2

i ii i

= cos2

0 1

x

edxx

log

1

1

1 k kk

...

... 1

.3834737192747465...

( ) ( )

1 2 11 2

1

k

k

k

21 !

log

2!

1

k

n

nn k

k

n

.38349878283711983503... 8 2 4 2 161

0

1

Gx x

x

log

log( ) log

.38350690717842253363... ( ) ( )2 2 11

k kk

1

.383576096602374569919...

1 43

4log

3

4

kkkH

= 7

18

.383917497444851630134...

12 16

122cot

24112

61

k kk

.383946831127345788643... 2

2

1

31

1

1

2

( ) ( )

( )

k

k kk

.38396378122983515859...

23 1k

k

k

H

... 11

2

2

2 11

21

2 1

2 11 1

kk

k

kk

kk

Q k( )

.384255520734072782182...

12 2

11

k k

.384373946213432915603... dxxx 1

0

)(log)2(sin2

2log

GR 6.443.1

.3843772320348969041... ( )4 1

4 414

1

k k

kk 1k k

= 1

8 2 2

1

2

1

2

i i

= 1

22 1

1

21

1

22

1

22

1

21HypPFQ , , , , ,

1

62 1

21

22

22

21HypPFQ

i i i i, , , , ,

.384418702702027416264...

0 )3)(1(3

1

3

2log12

4

21

kk kk

1 .384458393024340358836... dxxx

x

1

021

arctan)21log(

2

GR 4.531.12

... 1

3

2

3 6 12 62

2 2

40

4

G x

xdx

logcos

/

1 .384569170237742574585...

0

22

)1)(3(

log

8

3log

8dx

xx

x GR 4.232.3

... sin cos log( cos ) ( ) sin1

2

1

22 2 2 1 1

1

22

= sin( )

( )

k

k kk

1

11

GR 1.444.1

=5

13

.384654440535487702221... 3

2log

3

2

2

3log3log

2

2log3log2log)3( 2

322

Li

3

2

3

133 LiLi

=

1 1

)2(1

2 2

)1(

3k kk

kk

kk

k

H

k

H

7 .38483656489729528833...

3

2)2(

.38494647276779467738... log log2

4 2 220

x

x

.3849484951257119020... ek k k kk k

7

3

1

3

1

1 21 1!( )! ( )! !

... ( ) ( )/ / / / / /

1

1

44 2 2 3 1

1

44 2 2 32 3 1 3 1 3 1 3 1 3 1 3 i i

1

3 21

1

22 3 2 3/ /

= ( )3

4

1

4 1311

k

kkkk

.38499331087225178096... )1(!2

1)1(2

2

1

0

11

kkJ

kk

k

.385061651534227425527...

22 1

1

2

5arctan

3

2

3 xx

... 3

13

... 6

...

3

1

61 3 1 3

1

331

i ik kk

5 .38516480713450403125... 29

.38533394536693817834... log cos ( )cos

2 1 1 12

1

k

k

k

k

... 1)1()sin()1(222 1

1

kkeei

k

kii

... /1

... arccothee kk

k

1

2

1

2

1

2 12 10

log tanh( )

J945

= dx

e ex x

1

...

12

1

1 1

2)12()!1(

)24()1(

kk

k

kerf

kk

k

... 1

21

2

0

e xx ( )dx

... 2 2 11

2 1

1

2 2 401

log( ) (

k kkk k k )

= 1

4 121 ( )k kk

J374, K Ex. 110d

= ( )

( )

1

1

1

1

k

k k k J235, J616

= ( ) ( )

( )( )k k

kkk

kk

kk

1 1

2

2 1

41

1 21

k

= 2

1

11log

x

dx

x

= E kdk

k( )

10

1

GR 6.150.2

... 2 2 43

4

3

411

log log

Lik

k

kk

= ( )

( )

k

k k k

Hk

k kk

k

kk

k

1

2

1

2

1

2 1 212

1 0

1

=12 1

4 1

2

2 21

k

k kk

( ) J398

= 1 21

8 231

k kk

[Ramanujan] Berndt Ch. 2, 0.1

=

11 4

12

)!2(

!)!12(

kk

k kk

k

kk

k

=Hk

kk 21

= x x x

xdx

n n n

1 1 2 2 1

0

1 2

1

/

GR 3.272.1

=log( )1

21

x

xdx

= log 11

0

1

x

dx

= log( )

11

20

x xdx

...

1 2

2log22k

kkkH

... 27 2 22 1

4

1

log( )

k

kkk

... 925 2 22 1

6

1

log( )

k

kkk

.3863044024175697139... k

kk 4 11

...

12

1

kke

...

4 4 2

1

2

22

32coth coth csch csch

2

= ( ) ( ) ( )

1 2 21 2

1

k

k

k k k

... 1

4 4 5

5

2

1

4 421

tan

k kk 4

... log6 2

... Lie kk

k3 3

1

1 1

3

... G x

xdx

2

7

83

2

0

1

( )arctan

.38716135743706271230...

1

2

310

4)!(8

)1(

4

)1(

kkk

kII

.38779290180460559878...

125

1

5

21

24

5log5

55

55log

4

5

k kk

11 .38796388003128288749... 2

34 3

22

22

0

( ) ( )x Li x dx

...

1 )!2(

!!1

k k

kk

... 2

1arcsinh21log

2

1 22

=22

1

!)!12(

!)!2()1(

0

kk

k

k

k J143

= arcsin x

xdx

1 20

1

=

1

021

arcsinhdx

x

x

... 34

15 CFG D1

... ek

k

k

k

k

k

k

2

0 0

12 1

3

2 1

!

( )

! LY 6.40

... 6

...

123

1

2

4

1

4

)2(

622log12

kkk kk

k

.39027434400620220856...

1 )13(

11

k kk

.39053390439339657395...

0

12 )12(3!

12

3

1

kk kk

erfi

2 .39063... 1

22 ( )kk

...

0 1

2

),2(

),2(

k kkS

kkS

2 .39074611467649675415...

0 2

)3(222log8

kk

k

1 .39088575503593145117... log

2

3

.3910062461378671068... 64

22

0

1

arcsin arccosx x dx

...

12 544

1tanh

8 k kk

GR 1.422.2, J949

... 2 3 2

0

2

3

2

3 1

log log log

( )(

x

x x 2)

...

1

11

keke kee

Li

.39173373121460048563... sinh

cosh1

3

13 4

1

x

dx

x

.39177... 1

122

k s

ks

.391809...

2 )1(

)(

k kk

k

.391892330690243774136...

12

22

2

)1(22log2log4

62

kk k

k

... 2

2

04 2sech

x x

xdx

cos

sinh

.39207289814597337134... 16 2

22

1

( )

( )

... 4

2/3

1 .392096030269083389575... F kkk

2 12

2 1 1

( )

... 33

... e

e x x dxx4

1

0

2

log

.3925986596400406773... = 2 1 11

2 11

erfi Eik k kk

( ) ( )! ( )

...

1

4

!

44

k

k

k

ke

.392699081698724154808...

0 24

)1(

8 k

k

k

=

0 )34)(14(

1

k kk

=

1 02413

1

04

04 )1(44 x

dxx

xx

dx

x

xdx

x

dx

=

022 xx ee

dx

= dxx

xx

0

2 2sinsin

=

05

222 cossindx

x

xxx

= GR 4.523.2 1

0

arcsin dxxx

.3926990816987241548... = 12arctan8

= ( )

( )sin

1

2 1

2 1

2

1

21

k

k k

k J523

= ( ) ( )4 1 2

16

1

4 1

1

16 112

12

1

k

kk k k

k

k k

= 1

4 2 1 2 3

2

0k

k k k

k

k( )( )

= arctan( )

1

1 2 21

kk


=1

4 2 211 ( )s r

sr

J1122

= dx

x20 16

=

0

coslog dxxxex x

= x x dxarcsin0

1

4 .39283843336600648478...

1

4

8 )()4()4(4

2160 k k

krL

.393096190540936400082...

12

22

0 )!(

2)1()22(2

k

k

k

kJ

...

12

2

)12(4

2log

4

)3(7

k

k

k

H

... 11

71

k kk ( )

.393469340287366576396... e

e k

k

kk

1 1

2

1

1

( )

!

... 2

6

...

3 !

1

k kk

... 2 2 1 2

22

2 22

k

k k

k

kLi

j

( )

... )3(34

...

0 !

8)1(50

k

kk

ke

... coth

8

... 3log4

...

1

)3(1

2

)1(

kk

kk

k

H

... 1)2(14

3log

1

kk

k

k

.394784176043574344753... 25

2

2 .394833099273404716523...

01 )!1(!

222

2

1

k

k

kkI

.394934066848226436472...

3

2)1(

2 1)3,2(3

4

5

6 k k

= 11

)(2

122

i

eLieLi

= ( ) ( )( ( ) )

1 12

k 1k

k k

=

2 11

1)1(

j kk

k

j

k Berndt 5.8.4

=

134

logdx

xx

x

= dxx

xx

1

0

2

1

log

1 .39510551694656703221...

3

)(k

k

... 11

2

kk !

...

( )

( )cos/1

1 20

e

ee xx dx

... ( ) ( ) ( )( )( )1 1

1

5

8

7

8

3 1 1

82

k

kk

kk

...

1 )5(!44120

k kk

ke

...

0

3/1

3!

1

kkk

e

...

1

22

)!3(

2

8

19

8

3

k

k

k

ke

... 2/7)2(

2

311025

51664H

...

1

21

!

1)1()1(

k

k

k

k

1 .39689069308728222107... k

k

kk

2log

1

12

.39691199727321671968... 2 2 1 12

2 11

1

sin ( )( )!( )

kk

k

k

k k

.39710172090497736873... 1

2 3

2

31

1

3 121

csc sin

kk

J1064

.397117694981577169693...

eerf

1

...

12

1

0 )!(

4)1()4(1

k

kk

kJ

.39728466206792444388... 3 3

4 2

3 2

20

sin x

xdx

... dxe

xx

e x

1

2 sin1cos2

...

0 )42)(12(

1

3

2log

6

1

k kk

3 .397852285573806544200...

1

2

)!22(4

5

k k

ke

.3980506524443333859... | ( )|( ) k k

kk

1

2

1

1

... 4

3 8

... 1

0

)tanarctan(arc dxx

...

2 6

2

3

1

4

1

3 221

coth

kk

... 1

2

... 13

2 ( )

.3990209885941838469... 2

2 2 2 2 21

2

20

cos cos ( ) sin( ) ( )sin

( )

si cix

x

... 1

4 17

17

2

1

5 221

tan

k kk

... e

k k k

k

k

2

0

1

16

2

2 4

!( )( )

... ( )

12

1

1

k k

k

Hk

kk

= 2

51

21

21

1

1

1

( ) ( )k kk

k

k kk

k

F kF

= 1

2 21 2 2 1

0cosh log( ) ( log )( )

k

k

e k J943

= sin sin cos2

0

2

02

0

x

edx

x

edx

x

edxx x

x

= 12

5 42

1

F kkk

k

... 121

1)1(

kk

Hk

... log 2

... e G

... e Eik

k k k kk k

1 11

1

221

20

( )!( ) !( )

= e x x dxx log0

1

... ( )( )

arctan

12 1

1 1 1

2 1

k

k k

k

k k k k

...

1

)2(log)(k

kk

.400685634386531428467... ( ) cos( / )3

3

33

1

k

kk

... 4 2

12 2

4 22 2

cos sincosh sinh

=sin2

40 1

x dx

x

... 1

62 1

1 22

( )

( )

( )(

k

k kk )

Admchick-Srivastava 2.22

... ( ) ( )

!

//k k

k

e

ke

k

kk

k

11

2

11

1

.40148051389327864275... log( )

27

24

1

4

1

23

k

k

k

k

... 912 1

3 21 30

e k k

k

kk

/

( )

! ( )

.40175565098878960367... 5 2

12

3

16 1

5

0

4

cos

sin

/ x

xdx

1 .40176256381563326007... log

1

3

...

6

1

3

1

6

1

6

53

4

=

06

1

03 11 x

dxx

x

dx Seaborn p. 168

1 .402203300817018964126... 3

46

23

0

23

0

1

x x dx x dsin arccos/

x

... 3

4

2

...

3

4

2

42

1

222

csc

( )k

k k

.40268396295210902116... Li32 34 3

5

4

52

2

3

( )( ) log log

5 1

2

=5

)3(4

2

15log

15

2

2

15log

3

2 22

... k

k kk

!

11

... ( )

!!

k

kk

1

2

.40306652538538174458...

03 2739

2

x

dxx

... e erfk k

k

1 18

0

12

1

3 2

1

3/

!!

6 .40312423743284868649... 41

... 0 1 0 20

11 2 1

Fe

Ie k ek

k

; ;( !)

... 2

1

2

2

( )

( )!

k

kk

... 1

8

1

3 13

2 230

1

log

( )

x

x xdx GR 4.244.1

... 98

2431

21

6

1

( )kk

k

k

... ( )

1 1

3 31

k

k k k

... K x

xdx0

2

20

2 2

2 1

( ) log cos

...

0 02 )1()1(

)1(1xe

dxx

xe

dxEie

xx

... 2 11

3

0

eEix dx

e xx( )( )

.403936827882178320576... 1

2 121

k

k

... 1

3 1

1

3 11

1

1k

k

k

kk

( )

... 2 3 2 22 ( ) ( )( )

= log log

( )

2

3 21

2

0

1 2

01 1

x

x xdx

x x

xdx

x

e ex x

GR 4.26.12

2 .4041138063191885708...

1

2)2( )1()3(2

k

k

k

H

=log ( ) log2

0

1 2

0

2

21

1

1

x

xdx

x dx

e

x

x xdxx

=log2

0

1

1

x

x

= dydxxy

xy

1

0

1

0 1

)log(

... 2 3 19 2

1

4 2

2 32

( )( )

k k

k kk

1

= (( ) ( ) ( )2 1 2 2 12

1

k k kk

... 82

3 1 1

41

2

1

Gk

kk

kk

( )( )

( )

...

2

31

6

3

10

cothsin

x

edxx

... 2 2 1

2 1

1 22 1

1 2

k

k k

k

k k

( )

( )!sinh

k

.4045416129644767448...

2 )1(

1)()1(

k

k

k

k

.40455692812372438043...

csc ( )

2

1

2

1

1

1

21

k

k k

... k

k kk

2

1

... 11 1

1

Eie k ek

k ! k

... 9 182

23

63

6 iLi e Li ei i( )

=

sin63

1

k

kk

GR 1.443.5

...

2

1

21 1 1coth i i

= )12()2()1(1

1

1

13

kkkk

k

k

k

k

... cosh sinh

( )

2

2

2

2 2

2

2 11

k

k

k

k !

.40496447288461983657... ( )2

... cos(sin ) cosh(cos ) sinh(cos )cos

!2 2 2

2

0

k

kk

... ( )

11

12

51

k

k

k

k

... 3 1

4

1

2 1

1 2

1

cos ( )

( )

k

k

k

k !

... e /2

2

.40528473456935108578... 4

2

... log3

2

1

3

1

311

Likk

k

J102, J117

= ( ) ( )

( )

1

2

1

2 2 2

1

1 0

k

kk

k

kkk k

= 21

52

1

5 2 12 10

arctanh

kk k( )

K148

= dx

x x

dx

e

dx

ex x( )( )

log

1 2 1 20

3

01 1

.405577867597361189695...

2 15 3 520

dx

x

... H

kk

kk

( )

( )

2

1 2 1

... ( )

!

/k

k

e

kk

k

kk

2

2

1 2

210

... ( )( )

arctan

12

2 1

1 11

1 1

k

k k

k

k k

k

.40587121264167682182... 4 3

31240

log x

x xdx

... ( ) ( )2 3

.40638974856794906619... 2 27 3

2 1

2 2

20

1

G

x x

xdx

( ) arccos

... sum of row sums in 1

k kb a

.40671510196175469192... sinh

sinh2

4

1

22

0

1

x dx

1 .40671510196175469192... sinh

cosh2

4

1

22

0

1

x dx

... 5

1

2

1

2 5

3 2

10 1 5

2

2 5 1 5

5 5

20 1 5

5 5

5 5

arccsch arccsch

( ) ( ) (log

1

5 1 55 2 5 1

( )log log( )

= ( )

1

5 20

k

k k

2 .40744655479032851471... ( )

!/2

11

11

2k

ke k

kk

... 1

6 182

1

2 2

40

1

log

log

( )

x

xdx

... 6

6

.40825396283062856150... 11

21

kk k

.408333333333333333333 = 49

120 6

1

6

1

96

21

24

H

k k kk k

.40837751064739508288... 1

164

2 2 2

2 2

1

2

7 4 3 4 3 4

3 4 3 4 41

/ / /

/ /

sin( ) sinh( )

cosh( ) cos( )

kk

... ( )

1

1

1

41

k

k

k

k

... 14 5 e

.40864369354821208259... sinhcosh

1

2 81 1

12 4

1

erfi erfx

dx

x

.408754287348896269033...

1

2

2 )!1(1

6

1

k

k

kk

B

eLi


=

12

1

kk ke

... e

e dxx2

1

0

2

... 4 3 1

290 4

( ) ( )

=

442

)()3)(2)(1(

kk

kkkk

... 4 3961

2

( )

...

0 12

)3(93/1xe

dx

... e

x xe(sin cos )log sin log

1 1

23

1

dx

... 3

22

... ( ) ( )

1 2 11 2

1

k

k

k

... 2

317

a k

kk

( ) Titchmarsh 1.2.13

.410040836946731018563...

2 1)(

1)(

k k

k

.41019663926527455488...

1

)4(

3kk

k

k

H

... 21

F

k

k

... ( )/

1 11

21

kk

k

e

k

... 11

2

1

2 1

2

20

1

arctan

tan cos

cos

x

xdx

.4106780663853243866...

1

1

2!!

)1(

kk

k

k

...

100 2!

2

)!12(

2!

!)!12(

1

2

1

2 k

k

k

k

k

k

kk

k

k

kerf

e

2 .410686134642447997691...

0 0 )!2(

!)!2(

)!2(

2!

2

1

21

k k

k

k

k

k

kerf

e

... sine

... 3 93

4

1

3 20

log( )

( )

k

kk k

... ( )( ) ( )

11 1

2

k

k

k k

k

... 1

1 k kk !!

... 2

2124

2

4

1

4

( )

kk

= x dx

ex

3

0

2

1

= log logx

x xdx

x x

xdx3

12

0

1

1

= log log1 112

02

0

e dxx

dx

xx

= GR 4.384.12 log(sin ) tan/

x x d0

2

x

... 1

21 1 1 1 1

0

1

cos cosh sin sinh cos sinh x x dx

...

0

2 3/1

)1(

3csch

2

3

2

3

k

k

k

1 .41135828844117328698... SinhIntegral

k k kk

( )

!( )( )

2 1

2 1120

2 .411474127809772838513... 2 318

1 8 8 8 8cos ...


1 .41164138370300128346...

2 1

2

sinh (log ) sin ( log )i i

... 7G

= 7

17

... 11

131

kk

... 3 3

8

24

1

log log( )

x

xdx

... csclog2

0

11

x dx

x

.4126290491156953842... 1

2

1 1

12

( )

( )

k

kk

5 .4127967952491337838... kk

k

2

1 2 1

... ( )

(( )!)

2

2

1

2

2 212

10 0

k

kI

kJ

kk

... e5

.41318393563314491738...

( )4 1

4 11

kk

k

.41321800123301788416... 2

3

2 3

9

1

2 arcsinh

= 2 11

111

2

1

2

1 22

Fk

k

k k

k

, , ,( )

... 1

331 kk

.413292116101594336627... cos(log ) Re i

2 .4134342304287...

2 1

21

1n k

nk

... 2

3 3 5 52 3 30

/

dx

x

... 1

2 4

2

0

cos cosh( )!

k

k k

6 .41376731088081891269... 16

243 1

4 3

30

log x dx

x

... e

e x x dx

1

102

0

sin cos x

... log12

6 42 1

1

H kk

k

.4142135623730950488... 8

5cot

8tan

2)!2(

!)!12(12

1

kkk

k

... 2

= 1

8

2

0k

k

k

k

= 11

2 11

( )k

k k GR 0.261

.4142135623730950488... 8

csc8

3sin21

.414301138050208113547... 1 2

3

3

6 3

1 3

4

3 3

4

log

( )

i

i

i i

2

6 3

3 3

4

1 3

4

i

i

i i

( )

= ( )

1

1

1

31

k

k k

... 1

31 k

kk

.414398322117159997798...

13)2/1(

1

2

3,38)3(7

k k

8 .4143983221171599978... 7 31

12

31

( )( )

kk

... 11 1

21 1 1 1 1

eE i i i i( , ) ( ) ( ) ( , )

= ( )

!( )

1

1

1

21

k

k k k

... 7

6 2 6

3

2

1

2 322

cot

kk

.4146234176385755681... 4

3

2

9 31111

1

22

1

4

HypPFQ { , , , }, , ,

= k

k

kk

k 21

1

( )

.414682509851111660248... 1

2 pp prime

.41509273131087834417... 1)1(2

22log1

kk

k

k

... 2

e

.41517259238542094625...

1

)3(

3kk

k

k

H

... ok kj

k k jk( ) ( ) ( ) )

1 1 12 11

1

... 11

6 2 22 2 4 4

1

coth cot ( )k

k

k 1

= 4

442 kk

... 25

9 1

2

6 50

log

/

x dx

x

... k kk k k

kk

33 6 3

3 41

3 14 1

1 ( )

( )

( )

... 10

...

( )( )

( )2 1 1

3 36

122 4

22

22

11

f k

k jk kj k

... G E x 1

22

0

1

( )dx Adamchik (17)

.416146836547142386998...

0

1

)!2(

4)1(2cos

k

kk

k

1 .416146836547142386998... 2 1 1 2 14

22

1

sin cos ( )( )

1

!

kk

k k GR 1.412.1

7 .4161984870956629487... 55

...

1

4

1

4

1 1

42,

... arctan 3

3 220

dx

x x 10

.41652027545234683566... 3

16 2 2 12 30

dx

x( )

...

( )34

.416666666666666666666 = 5

12

1

222

k kk

...

( )2 1

4 11

kk

k

.41715698381931361266... | ( )| k

kk 4 11

... 7 3

2 2 84

32

27

1 1

4

2 3

12

( ) ( )( ( ) )

Gk k k

kk

.417418352315624897763... 3

81

4 20

erfi

e k

k kk

!( )3

.417479344942600488698...

1 )2(

)4(Im

kki

k

.41752278908800767414...

k

k

k

k

kk 2

)2()1(

)12(

112log4)2( 1

12

.417771379105166750403...

0

4

222 cossin

23

1dx

x

xxx GR 3.852.5

.41782442413236052667...

1

330 26

)1(

6

2log

36 xx

dx

kk

k

... root of ( )x 3

... 11

1

2

1

1

2 1 1 21

e

e

e ekk

k/

=

1 )1(

11

k

k

kke

e GR 1.513.5

.418155449141321676689... )(Im i

.41835932862021769327... 1

3293 5 7 2

2 2

0

1

( )arcsin arccos

x x

xdx

2 .418399152312290467459... 4

3 3

3

2 2 11

k

k

k

kk

!

( )!! J278

= area of unit equilateral triangle CFG G13

=

1

21

21

)!12(

)!()!(

k k

kk

= dx

x x

dx

x20

3 201 1

/

= dx

e ex x

1

.41868043356886331255... 35

2592

5 63

31

sin( / )k

kk

GR 1.443.5

1 .419290313672675237953...

3

3

2 2 3

1 3

2

1 3

2

2 2

tanh

i i i

i i i

2 3

1 3

2

1 2

21 1 ( ) ( )

= H

k kk

k2

1 1

97 .41935701625712157163... 4 4

.41942244179510759771... 11

2 11

kk

... e

e e

k

ekk

sin

cos

sin1

1 2 121

... sec(log )

2i i

... 1

4

1

2

2

4

1

2

1

2 1 2

1

1

cos sin( )

( )!

k

kk

k

k

148 .41989704957568888821... e e5 5

... 92

3

1

1 3

1

1 3

2 1

2

1

21

( )

( / )

( )

( / )

k k

k k k

1 ...

1

3

)(2 2

)6(

)3(

k

k

k

HW Thm. 301

=

primep p

p3

3

1

1

... 2

0

2

2

loglog sin x x dx GR 4.322.1

...

0 )3)(1(2

12log3

2

5

kk kk

... ( )!

1 11

1

kk

k

k

k

... 1

532 kk

...

8 21

2

30

log

e x

edx

x

x GR 3.455.1

.42073549240394825333... sin ( )

( )!

1

2

1

2

1

1

k

k

k

k

...

02

2/

0

01

cossin)sin(tan)1( dx

x

xdxxxK

= 8

19

...

1 1

0

1 144

14

4

)(

14

12

k kkk

k

kk

k

k

6 .42147960099874507241... e

e

e

k

k

1

.421643058395846980882... ( )( )

12 1

1 0

1

kk

k

k

.421704954651047288875... 2 21

2 220

Jk k

k

kk

( )( )

!( )!

... 3 2

2

9 3

16

6

30

log log sin

x

x GR 3.827.13

.42176154823190614057... e2 2 2 2 2 2 2 2 2 2cos sin( sin ) sin( sin ) cosh( cos ) sinh( cos )

=

ie e

k

ke e

k

k

i i

2

2 22 2

1

2 2 sin

!

... k H

kk

kk

2

1 2 2 1( )

... sin51

... 1

21 1n

kn k

n

( )

... H

kk

k

2

31 1( )

1 .4220286795392755555... log log

( )3 2 2

3

2

3 6

2 2

34

1

23 3

Li

=x x

x xdx

log

( )( )

2

21 2 1

...

( )

( ) ( )

3

2 3

...

2

1122

1

2

log( )k

kkk

...

33 1

3 1

1

1 60

log( )

( /

k

k k )

... 1212log4

1

16 22

2

Berndt Ch. 9

... 11

3

1

2

2

2

1

1

( )k

kk

k

...

2

1)()2(1

k k

k K Ex. 124g

=

221

1

1log

11

kk kk

k

kkLi

=

0

logdx

e

xxx

= x

dx

xx

x

0 1

1sin GR 3.781.1

= cos xe x

dx

x

1

10

1 .42281633036073335575...

122

1k

k

k

.422877820191443979792...

5

23Li

.422980828774864995699...

1 )2()!2(

4)1(2log)2(

k

kk

kklCosIntegra AS 5.2.16

.423035525761313159742...

1

21)2(k

k

.423057840790268613217...

012 )23(2

)1(

2

1,

3

5,

3

2,1

2

1

kk

k

kF

.4232652661789581641... ( )

( )3

360

4

31

k

kk

... e

... 2

)3()2(

...

2

12

12

1

2

1

4

12log

iiii

=

13

1)1(

k

k

kk

.423565396298333635... B

kk

k ( ) !

20

.423580847909971087752...

1 2

1)1(

kk

k kH

2 .423641733185364535425...

0

32

)!3(

2

3

3cos2

3 k

k

ke

e J803

.423688222292458284009...

0 )4(2

1

3

322log16

kk k

.423691008343306587163... 1

0

2sinlog22log2

1dxxxlCosIntegra GR 4.381.1

.423793303250788679449...

3

2

1

7

)1(

21

57csc

72 k

k

k

.424114777686246519671... 1

0

32 log)1log()3(6)4(624 dxxx

.424235324155305999319...

2

1)(k

kk

1 .424248316072850947362...

12

12

1 11)1()1(

kk

k

kLi

kk

k

.424413181578387562050...

2/1

1

.4244363835020222959...

2

1

2

121 4

1

1 0

2

eerfi

kk

k

e xk

k

x/

( )

!

sin

dx

.424563592286456286428...

02 15

1

15

7

2

21tan

21 k kk

.424632181169048350655...

2

1)( 1)(k

kk

14 .424682837915131424797... 12 31

2

20

( ) /

x dx

ex

.42468496455270619583...

22 1

2

20

loglog sinx x

xdx

1 .424741778429980889761...

12

1

1 1arctan

12

)24()1(

kk

k

kk

k

=

2

11log

2

11log

2

11log

2

11log

2

iiiii

...

328 3 123 2csc cosh cosh sinh coshh

= ( ) ( )( )

( )

1 2 11

11 2

1

2 2

2 32

k

k k

k kk k

k

9 .424777960769379715388... 3

.425140595885442408977...

2

31

2

3122

3

2 ii

=

1

1 )3()13()1(k

k kk

.425168331587636328439... dxx

x

e

22 1

2cos

.4252949531584225265... ( ) log arctan

12

1

3

3

22

1

21 2

1

k kk

k

H

... 2

coth

...

13

2

42log22log2

k

k

kk

H

.4254590641196607726... csch1

2 1

12 2

0

1

e

e e kk

GR1.232.3, J942

.4255110379935434188... B

k

k

kk

k !

2

0

2 .42562246107371717509... 2

3

7

42 3 2

3 3

1

2 2

32

1

( )( )

k k

k kk

= ( ) ( )( ( ) )

1 12

k 1k

k k k

.425661389276834960423...

1

)2(

3kk

k

k

H

.425803019186545577065...

0 )33)(13(

1

4

3log

312 k kk

.42606171969698205985... ( )( )

12 1

2

1

213

1

kk

k k

k

k k

=

1

21

21

2

i i

...

0 2)!2(

)1(2

4

kk

k

ke

8 .4261497731763586306... 71 ... 1)()1( xofroot

... 1

21 Fkk

...

2

53

2

15log

15 22

2

Li

Berndt Ch. 9

...

0

1

27

)1(

k k

...

1 )32(

1

3

2log2

9

8

k kk J265

1 .426819442923313294753...

G xx

dx

x

3 3

81 1

12

0

( )log( ) log

... 2

22 28

23

2

4 1

2 2 1

1

2

log( )

( ( ) )k

k k

k k

kk

k

... 12log22

...

1

21

)!2(

2)1(

4

2sin22cos2

k

kk

k

k

... dxx

x

4/

0

4

sin1

cos

812

12

... 3

2 21

3

1

21

log( )k

k k k

= dxxx

x

02

2

1

1log

= log( )

11

1 30

xdx

...

1

23

23

2

2 3cos)1(

2

1

124

9

k

kii

k

keLieLi

GR 1.443.4

...

0 22

)1(

2

1

2

22

k

k

k

... x x dtan0

1

x

2 .428189792098870328736...

133/23/1

11

)1()1(

1

2

3cosh

1

k k

Berndt 2.12

=

12

11

k kk

.42821977341382775376... eie

Re2

cos

.42828486873452133902... 1

log!

1

2

k

k

kk

.428504222062849638527...

1 )1(

)1)1)(log(1(

kk

k

e

H

e

ee

.428571428571428571 = 7

3

.4287201581256108127...

11

1

1

1

1

1

1 2eEi

k k k k

k

k

k

k

( )( )

( )!

( )

!( )

=( )

( )! !

1

1 22

k

k k k

... 6e

.42881411674498516061... 2 2

20

1

12

1

2

2

2

2

2

1

1

log log log( )

( )

x

x x

...

2

3

21

1

1 2

0

x

e edx

x x

/

.42909396011806176818...

0

3

)!3(

3

54

172

k

k

k

e

... 23

2

... ( ) ( )1 2 1 2 i i

... 22

=1

4

12 1

43

201 k k

k

kk

( )

=( ) ( )!

( )!

1 1 12

121

k

k

k

k

= log ( ) ( )2 11 2

21

1

k

k

k

kk GR 8.373

=2

22 1

2

2 1 2 2 11 1

k

kk

kk

kk k

k

k k k

( )

( )!!

( )!! ( )

=

sin 4

1

k

kk

= arccos arcsinx x0

1

dx

=

02 cosh)1( xx

dx

GR 3.522

.42926856072611099995...

0 1

log

3

13

2

2log3 xx ee

x GR 4.332.2

...

163

22

22

2 22 2csc cot cot csc

2

= kk k k

kkk

k

2

1

2 2

2 31

2

2

2 2 1

2 1

( ) ( )

( )

4 .429468097185688596497... 7

3

.429514620607979544321...

052 )1(256

35

x

dx

.429622300326056192750...

12

1

2

)1(

kk

kk

k

H

...

23

23

221)(2

kk

k

kkLi

k

k

... G

...

22

1 3

31)2(33cot

2

32

kk

k

kk

... dxx

x

04

22

)1(

log

9

3

...

1 1222 4

1

4

)2(

k kk k

Lik

k

.43026258785476468625... 1

2 1

1 3

231

1

1k

k

k

k

kk

( ) ( )

... 2

5

1

2

2

5

1 5

2

12

1

1

arcsinh

log( )k

k k

kk

=

0

122!)!12(

!)!2()1(

kk

k

k

k J142

=

1 2kk

k

k

F

=

0123 3xx ee

dx

xxx

dx

.43055555555555555555 =

1 )4)(3(

2

72

31

k kkk

k

.430676558073393050670... 2log5

11 .430937592879539094163... 48 64 211 2

2 20

log ( )/x Li x Lix

dx

... 5/16

...

1

0

1

0

1

022212

2log3

4

33 dzdydx

zyx

zyxG

2 .43170840741610651465... 24 4

22

( )

.431739980620354737662... 1

2 6 21 2

2

2

coth ( ) ( ) ( )k

k

k k

= k

k k kk

3

2 22

1

1 1

( )( )

5 .43175383721778681406...

1k k

k

F

H

.43182380450040305811... 1

5

1

2

1

2

1

2 2 1

1

2 11

arctan( )

( )

k

kk

k

k

.4323323583816936541... 1

2

1

2

1 2

103

12

0

1

e k

dx

x

dx

e

k k

k

e

x

( )

( )!

.43233438520367637377... 1

2

1 4

23 11

1

1k k

k 1

k

k

kk

( ) ( )

... ( ) log ( )

1 21

1

k

k

k

1 .432611111111111111111 = 4)2(

144

205H

... 2e

...

0

2

sin

log1

1x

x

...

1

0

1arctan

44arctan

xx ee

dx

ee

...

0

4

4

3

x

dxee xx

GR 3.469.2

...

2

3 1)()3(12)4(6)2(71k

kk

=

24

2

)1(

)14(

k k

kkk

... 81 3

2

189

4 1

2

1 30

1 ( ) log/

x

xdx

= 13

30

1 3

31

arctan /x

xdx

... k

kk

k

2

2 11

( )

... 413

... 1coshlog2

1log

2

e

e

=

ii

ii

21

21

11log

= 2 2 1

2

2 1 22

1

k kk

k

B

k k

( )

( )!

= 1

0

tanh dxx

...

1

2)2(

21

2sin2

k k

=

1 )2(

11

k kk

9 .4339811320566038113... 89

...

1 )7(

11

k kk

... 1)2(1)1(1

kkk

... log3

...

1

34/1

)!2(

!

2

1

128

79

64

45

k k

kkerf

e

... 10log

1log10 e

...

1

2

1

13

)1(

2

1

3csc

32 k

k

k

...

1 2

1

kk k

... ( )2 1

2 11

kk

k

...

01

2

0

2

)!2(

12)2(

6

5)2(

3

2

k kk

kI

eI

e

... 2 21 2 1 2 2 1

0

2 1sinh / / ( )!

k k

k

.4353670915862575299... 1

21 ( )!! !!k kk

... sin cos ( )

( )

2

4

2

2

1 4

2 11

!

k k

k

k

k

...

1

5

2!

)1(

32

23

kk

k

k

k

e

...

1

23

12 )1(2log2

121

k

k

kk

...

1

0 1

cosh

2

12log)1log( dx

e

x

ee

x

...

12

1)1()1(

)3/1(

)1(

6

1

3

2

4

19

k

k

k

.43646265045727820454... 2

22 2

125

2

5

1

5 2

1

5 3csc

( ) ( )

k kk

...

1

2

)!2(52

k k

ke

...

01

2

!

1

!2

kk k

k

k

ke

...

1

2

)2(1

2)1(

kk

kk

k

H

...

2 12

12

12

21

11

log ( )( )

i i

kk

kk

k

= 2 11

2 21

log

kk

1 .43674636688368094636...

3

1

2

3log3log2log

6)2( 2

22

2 LiLi

= log

( )log

2

20

1

02 11 2

x

xdx e dxx

...

22 2

1

4

3

2

7tanh

7 k kk

... 42 18 3

241

12

0

1

2

( )log logx

xx dx

...

2

)3()(3)2()4(k

kkk

=

24

23

)1(

12

k kk

kkk

.437528726844938330438...

4

2 14

)1(

1820

257

142

14csc

k

k

k

...

1

2

!

1)1(

k k

k

...

1

1

!

)1(1)(

k

kkk

e

k

HeeEie

... x

xdx

3

30 cosh

...

logx

edxx 10

1

...

0 )22)(12(

)1(

2

2log

4 k

k

kk

=

1 12

)1(

2

)1(1

k

kk

kk J72

= 1

0

4/

02

arctancos

dxxx

dxx

.43892130408682951019...

coth

2

1

2

1

121

kk

...

1

21)1(

k k

k

...

1 )2(

)3(Im

kki

k

...

2

2

1

2

1)(2

12)1(

)12)(1(

2

3

72log4

kk

kk

k

kkk

5 .43937804598647351638... 7

120

2

4

2

83

1

2 2

4 2 2 4

4

3

21

log log log

Lix dx

x x

...

1

1

2

2

)1(

kk

k

.43944231130091681369... e

e x

dx

x2

5

2

14

1

cosh

... 4 7 6 2 5 3 41

61

( ) ( ) ( ) ( ) ( ) ( )

( )

k

kk

... /1

.440018745070821693886...

0 )13)(2(

)1(

5

2log2

5

1

35 k

k

kk

.44005058574493351596...

0

1 4)!1(!

)1(

2

1)1(

kk

k

kkJ

.44014142091505317044... 2

12sin

1

1

2

1cos)1(

2

1sin)21cos22log(

2

1

1

k

kk

...

1

)22(Im

2tanh

2 kki

k

=

0 sinh

sindx

x

x GR 3.981.1

=

dxee

xxx

sin

2 .44068364104948817218...

02)1(!

1)(1

k

k

kk

eeEi

e

...

12

1

0 2)!(

)1()2(1

kk

k

kJ

.440993278190278937096...

2

3

24

3

Berndt 2.5.15

3 .441070518095074928477...

1

3 1)3(9)2(12)3(6)4(6k

kk

=

242

2

)1(

14

k kk

kk

3 .4412853869452228944...

3

4

23

4

1

4

3

8

5 4 3 4/ /

sin

... 31

13

230

( )( )k kk

= dxx

x

02

2 )3log(

3 .441523869125335258... e Ik

k

kkk

00

11

2

2( )

!

... )4(2)3(3

... )572161log(16

52log2arccsch

4

5

10

55

=

0 5/1

)1(

k

k

k

.44195697563153630635... ( ) ( ( ) ) log

1 1 11

11

1

1 2

kk

k k

H kk k

... 3/13

...

... ( )

! ( )

1

1

1

1

k

k k k k

.44269504088896340736... 1

21

1

2 1 2

2 2

3 1 2 21 21

1 3

1 3 2 31log /

/

/ /

kk

kk

k

k

k k


1 .44269504088896340736... 1

2log

...

1 2

3)2()()1(

)1()3()2(

k k

k kkk

k

=

12)1(k

k

kk

H

.44288293815836624702... dxx

x

1

0 1

arcsin42

4 .44288293815836624702... 21

4

3

4

=

dx

e

ex

x

1

4/

= log log1 14

0

2

0

x dx x dx2

= dxx

x

02

2 )2log(

=

x

dx2sin1

...

1

21221

)!2(

)12(2)1(1tanlog

k

kkkk

kk

B AS 4.3.73

...

00

2 42

4dxxedxex xx

LY 6.30

= dxxx

xxe x

2/

04

2tan

cotcos

cos12

GR 3.964.2

.443147180559945309417...

2

2 1

)1(

4

12log

k

k

k

k

.443189592299379917447...

12 24

12cot

224

3

k kk

= 1)()1(2

2

kL

kk

k

1 .443207778482785721465...

1

2

)12(

)1(2log2)3(2

k

k

kk

Hk

.443259803921568627450 = )15(8160

3617

.443409441985036954329...

0

2/)12(2/12/1

)1(2/1cosh2

11

k

kk eee

J943

3 .443556031041619830543...

12!

2

4,2,2,2,2,2

3,1,1,12

k kk

k

k

HypPFQ

1 .443635475178810342493... 2arcsinh

.4438420791177483629...

log ( )( )

!2

1

2

1

2

1

1

Eik k

k

kk

... 25

.444444444444444444444 =

1 49

4

kk

k

... , maximum value of ee /1 xx /1

=

0 !

1

kkek

5 .4448744564853177341...

32 3

022 3

( )log xdx

ex

... ( ) ( )( )

( )

1 12

8

2 12

2

31

kk

k k

k kk k

k

1 .44494079843363423391... 2 03

1

3( )( )

k

kk

Titchmarsh 1.2.1

...

( )

( )

2

2 11

k

kk

... 9

3

... 32 3

3 3

2

1

3 21

log

k kk

=

4

3

1

31

1

31

1

hgkk

kk

( )( )

= 1

0

3)1log( dxx

.445260043082768754407...

1

2

65

6log

125

42

125

48

kk

kHk

...

1 3

)1(

2

3log1

4

3

kk

kHk

... )()(2

1)4()2(2

3

2

3

1 24

24

ii eLieLi

=

1

4

2cos

k k

k GR 1.443.6

...

0 )2()!12(

)1()21sin21(cos2

k

k

kk

.446577031296941135508... ( )

( !)

k

kI

k kk k2

20

1

21

1

...

2 )(

)1(1

k k

k

2 .4468859331875220877... k

dkk

1

2

... 2sin22cos2

=

1

1

33 )1()!2(

2)1()()(

k

kkii

kkeLieLi

.447213595499957939282... 5

5

... 128

27

32

3

4

3

24

27

144 2

27

2

G log

=log( ) log

/

11 4

0

1

x x

xdx

.44745157639460216262... 3 3

2 21 2

1 33 3

0

/

( )(

dx

x x )

...

2

23 2

112

3

8

1

21 1

1

2

( ) ( )k

k k

k kk k

2 .4475807362336582311...

2

3

21

32

3

2 3( ) /

.44764479114467781038... 2

3tan

.44784366244327440446... ( )

( )log

1

3 11

1

31 1

k

kk

kkk

...

12 2 61

6

3

2 125 6coth csc ( ) tanh/

i icsch

2

= ( )

1 1

4 21

k

k k k

810 .44813687055627345203... 2sinh2coshsinh2sinh2coshcosh2

1

2

2

eeee

=

0 !

cosh

k

k

k

ke

... 4 2

12 1

2 1 20

cos( )

( )

k

kk

GR 1.444.6

.44841420692364620244... log

log loglog2 2

2 2

2

22 3

3

2

1

3

1

2

Li Li

=

3

263log32log3

6

12

222 Li

=

112

1

2

22

32

)1(

4

1

2

1

2

2log

12 kk

k

kk

k

k

H

kLi

= ( )

( )(

1

2 1 20

k

kk k k )2

=

12

21

02

2

)12(

log

)2(

log

x

xdx

x

x

= log 120

e

dxx

.44851669227070827912...

1

22

2

1

2

)15(

5

5

)2(

5csc

5

2sin52

40 kkk k

kkk

1 .448526461630241240991...

122

12

1 1)2()1(

kk

k

kLi

k

k

...

1 !

1

k kHk

...

1

333

cos

2

1

k

ii

k

keLieLi

...

0

3/1 3)!2(

)1(6

9

kk

k

ke

... 7 1

5 2

70

x dx

x

/

... 32

9 3 2 2

12

12

1

( )!(

( )!

k k

k

)!

k

.449282974471281664465... Li2

2

5

...

1322

1

2

1

x

dx

.449463586938675772614...

18 2

2

9 3

2

9 13 20

log

( )(

dx

x x 2)

... 6

... 41 11

6

1

ek

kk

( )!

...

12 2/14

1

22cot

221

k k

3 .4499650135236733653... 9 6 23

24 32

22

2 3

20

( )logx xd

e x

x

= 20

9

... dxx

xxx

x

x2

0

222

)2(

2log)1(log

3

2log

GR 4.313.7

... log( )

( )(2

3

3 2

1

2 1 3 20

k

k k k )

2 .45038314198839886734...

1

1

!!

2)1(

k

k

k

... 1

2 22 2 2

2 2

2

cosh sin sinh cos

cos cosh

= ( )

1

1

1

41

k

k k

.45066109396293091334... 3 2

2

2 1

2 120

cos sin ( )

( )!( )

k

k k k

...

13

1

2

)1(

8

)3(3

k

k

k

=

022

3

2xx ee

dxx

= 2 3 3 22

0

1

Li i Li i Li xdx

x( ) ( ) ( )

... )2(216

36

.45091498545516623899...

( )2

21

k

kkk

.45100662426809780655...

3

3

0

1

83 6 arcsin x dx

.45137264647546680565...

0

3

3

2

3

1xe

dxx

3 .451392295223202661434... loglog( )

34

1

2

20

x

xdx

.45158270528945486473... log( )

log 2

2

41

1

412

1

k

k kkk k

Wilton

Messenger Math. 52 (1922-1923) 90-93

= iloglogRe

= x

dx

x

x

log1

11

0

GR 4.267.1

= dxxx

2/

0

cot1

GR 3.788

1 .451859777032415201064...

0 !

sinhsin

4

1111

k

eeee

k

kkeeee

i iiii

...

0 212

)1(

13

13log

34

1

12 k

k

k

.4520569031595942854... 4

3)3(

... pk

kk

p prime k

2

1

2

( )

log ( ) Titchmarsh 1.6.1

.45231049760682491399... 3

4 2

3

2

2

3

1

2 3

1

21

csch

( )

/

k

k k

.452404030972285668128... 1

42 4 2

1

coth ( ) ( ) ( ) ( )

i i k kk

4 1

16 .452627765507230224736... 2

22

2 1

2 1

0

erfik k

k

k

!( )

... 1

6

1

2

1

3

1

2

5

6

1

3 20

, ,( )

k

k k

...

1

1)3(

)2(

k k

k

... 22

22log

28

5

2

2log5

28

5

16

51

=

2 8

)(5)1(

kk

kk k

... log2

3 20

1

e e

edx

x

x

... 1

2 ii

... 3 11

2

( )k

k

.453018350450290206718... 2

4

22

0

sin /x

ex

... e

6

9 .4530872048294188123... 16

3

42 2

40

sin ( )x

x GR 3.852.3

...

0

2 3

sin

3log1

1x

x

.453449841058554462649...

4 3 3 4 4 320

21

40

dx

x

dx

x

x dx

x 3

... 4 2

3

1

3 6

3 3

4

3 3

4

logcot

( )cot

(

i i

= ( )

1 1

41

k

k k k

.453592370000000000000 = pounds/kilogram

...

cot log log7

88 2 2 2

2 2

2 2

= 1

4 2 221

3 42k k

k

kk

k

/

( )

.45383715497509933263... sin/

x dx0

4

.45383998041715262... ( ) log ( )

112

n

kn

nk

... cot(log )

i i

i ii i

... 3

4

5

21

1

3 2

0

x

e edxx x

/

.454219904863173579921... Eik kk

k

1

22

1

21

log!

AS 5.1.10

4 .45427204221056257189... k

k kk k

2

21

21

4

21

2

2

.454313133585223101836... ( ) ( )

1 2 11

2

k k

k

... ( )kk

12

...

( )

( )

3 1

4 1

.45454545454545454545 = 5

11 63 2

1

F kk

k

... 2 3 21

21

( ) ( )

( )

k

kk

... sin

( )( )!

2

21

2

2 11

42

02 2

1

kk

k kk k GR 1.431

= 0

2 /

.454822555520437524662... 6 8 22 2

1

1 21

log( ) (

k

k Bkk k )k

... ( )

( )!

/2

11

1

21

2

k

k

e

kk

k

k

... 1

9log

... 22

21

7 31

( )log

( )

x

x xdx

... i

i ii

i4

3

22

3

22

sinh( ) ( )

= e x

edx

x

x

sin

10

.455698463397600878... ( )

logk

k k

k

kkk k

1

4

1 4

4 11 1

.45593812776599623677... i e i / /2 4

.4559418900357336741... 12

sinh

... 9

2

12

q2 q squarefree with an odd number of factors

Berndt 5.30

.456037132019405391328... 1

8

1

4

1

44

1

2i

ii

i itanh tan csc

1

84

1

2

1

4

1

4csc cot cothh

ii

i i

= e x

edx

x

x

sin

1 2

979 .456127973477901989070... sinh

2

2

2

21

1

kk

...

1

2

3

21

3

2 1 21

cos( )

kk

.45642677540284159689...

2 3 3

1

2

1

3 11

2

321

1

1

coth ( )( )

k

k

k

kk

k

... 1

2 2 3 3

1

3 120

coth

kk

= 137

300 5

1

55

21

H

k kk

.45694256247763966111...

( )

( )

2

3 1

1

312

1

kk

k k

k

.457106781186547524401... 1

2

1

4 1

3

0

4

cos

sin

/ x

xdx

... 2 21

2

1

log( )

!

!

k

k kk

... cot 2

.45774694347424732856... log2 1

3

1

12

3 3

2

3 3

2

3

2

3

2

= ( )

1

332

k

k k k

.457970097201163304227... 12

cosh

... G

2

= Adamchik (5), (6) log( sin ) log( cos )/ /

2 20

4

0

4

x dx x dx

= x dx

e ex x2 20

.458481560915179799711...

2 1

4 2112 2

1

2

1

sinh

( )k

k k k

.45860108943638148692... 7

1920

1

4

1

16

4

42

42

44

44

Li e Li e Li e Li ei i i( ) ( ( ) ( )i

= ( )sin

1 14

41

k

k

k

k

... 7 3

42 1

1

21

3

( )( ) ( )

( ) (

kk

k

k k k )

.458675145387081891021... 1 11

11 0

log( )ee k

dx

ekk

x J157

=

1 !k

k

kk


2 .458795032289282779606... 6 4 12 3 7 29

81 13

2

( ) ( ) ( ) ( ) ( )

k

k

k k

= 8 5 4

1

3 2

42

k k k

k kk

( )

1

2 .45879503228928277961... 7 2 12 3 6 49

81 13

2

( ) ( ) ( ) ( ) ( ( ) )

k

k

k k

= 8 5 4

1

3 2

42

k k k

k k

1

k

( )

... cos( sin ) (coshcos sinhcos )1 1 1 1

= ee e

k

k

ii e e

k

i i

2 1

2

1

cos

( )!

22 .459157718361045473427... Not known to be transcendental e

... 1

84

2

2 2

2

22

2 0

1

K

K

E xdx

x

( ) GR 6.151

... 7 6

45

3

2

12

51

( )( )

( ) ( )

k

kk

... 1

2

1

2

1

2 1 2

1

1

sin( )

( )!

k

kk k

.4596976941318602826... 21

21 1

1

22

1

1

sin cos( )

( )!

k

k k GR 1.412.1

= sinsin log

x dxx

xdx

e

0

1

1

= sin1

21 x

dx

x

...

1

... kk

kk

( )

( )

1

12

...

312

6 21

3

2

3 21

log( )k

k k k

... 2224

22

=

022 )2)(1( xx

dx

= dxx

xdx

x

x

2/

02

22/

02

2

cos1

cos

sin1

sin

... 9 e

... ( )k

k kk

2

... ( )

( )!!

1

2 1 22 10

k

kk k

...

1

2

... Gx

xdx

log log( )2

4

1

120

= log x

xdx

2

21

1

1

GR 4.295.14

= x

e edxx x

2 20

=

cos sin

cos sin

/ x x

x xdx

0

2

GR 3.796.1

= arccot x

xdx

10

GR 4.531.2

= dxx

1

02x-1

arcsinh

= GR 4.227.10 log tan/

10

2

x dx

1 .46057828082424381571...

2

1

77 22

0

arctandx

x x

... tan

!

k

kk

1

... 1

211

3

22

1

4

1 1

2

1

1

HypPFQk

k

k

k

, , , ,( ) ( )!

( )!

... 1

31 1

3 3

21 1

3 3

25 31 3

1 6 2 32 3

1 6 2 3

//

/ //

/ /

( ) ( )

i i

= 1

31

3 1

32 11

1

1k k

k

k

kk

k

( )( )

1 .46103684685569942365... tanh

!

k

kk

1

.46119836233472404263... )12sin(4)!(

)!2()1(

2

1sin

1cos2

1

02

kk

k

kk

k

Berndt 9.4.19

...

1

1

11

41

4

4

1

k

j

kk j

k

jlim( !)

.461392167549233569697... 3

4 2

1 1

2 120

cos

( )

x

x x

dx

x GR 3.783.1

=

0

5 log2

dxxex x

...

eEi

2

1

2

1 .4615009763662596165... 2

12

( )

( )

k

kk

.46178881838605297087... 9 2 3 12 2

301

14

0

1

3

loglogx

xdx

.461939766255643378064... 2 2

4

... 2 2

3

1

3 1 3 20

log ( )

( )(

k

k k k )

= dxx

x

3

1

0

2 11log

= dxx

x

1

04

6 )1log(

... e

e

B

k

kk

k

1

1

1

22

2 1

2

22

1

tanh( )

( )! AS 4.5.64

...

... 4

27

8 2

9

2

2

G

( ) J309

=

1

22 )23(

1

)13(

1

k kk

=log x dx

x30

1

1

1 .46243121900695980136... 11

2

p

p prime

... ( )

!!

k

kk

2

... ( )/

kk

k

11

2

1

2

1 .4626517459071816088... =

21

1

2 10

erfik kk

!( )

3 .4627466194550636115... 11

2 1

1

1 21 1

kk

kk ( )

... 205

44

1

4 10

log( )

( )(

k

kk k k 2)

... 1

21 2 2 csch

=

1

1 )22()2()1(k

k kkk

= )1(Re)1(

)1( )1(

222

ik

kk

k

= dxee

xxdx

e

xxxxx

00 1

cos

1

cos

... 3

6

...

4/1

4/34/14/3

3

1cot

3

1cot

3

1

4

i

=

1

223

1

k kk

=

k

k k

3

)24(1 1

... 21

22

1 5

2

12

21

21

arcsinh2

log( )k

k k

kk

... 6

.4634219926631045735... e Ei Eidx

e xx( ) ( )( )

2 1

10

1

...

1

8

85764801

319735800

kk

k

.463518463518463518 = 6

13

= 5)3(

3600

5269H

... arctan arcsin1

2

1

5

= Im log( ) Re log2 12

i ii

=( )

( )

1

2 2 12 10

k

kk k

=( )!!

( )!!

2

2 1 51

k

k kkk

= arctan( )

2

2 3 20 kk


= ( ) arctan( / )

12

11

21

k

k k [Ramanujan] Berndt Ch. 2, Eq. 7.5

.4638805552552772268... e

e

ek

k

k

( )1 1

1

... 3212

...

1

1

1 1

)1(

1

1

kk

k

kk ee

Berndt 6.14.1

...

23 6

)1(

k

k

k

...

2

2 )3()()1()3(4)4()2(9k

k kkk

...

1 )4(!249

k kk

ke

... e9 ... 3/1

.4646018366025516904... 5

4

AMM 101,8 p. 732

.46481394722935684757...

22 1

1

3

4

2

3tanh

3

3

k kk

=

2

35

2

35

3

3 iii

=

1

)3()13(k

kk

... 21

2

3

2

1

2 2 1 2 20

arctan log( )

( )(

k

kk k k )

= log( )

11

2 1 20

xdx

1 .465052383336634877609... 2

2

0

21

1

4 21

sinh

/

i kk

.46520414169536317620...

1

3/1

3!3 kkk

ke

.4653001813290246588...

2

21)(k

k

1 .4653001813290246588...

2

2 )()(k

kk

2 .4653001813290246588... 2

2

1( )kk

... 2 23 3

4 121

12 1

4 31

log( ) ( )

k

k k k

... 5 2log

.4657596075936404365... I

e k

k

k

k

kk

0

0

1 1

2

2( ) ( )

!

... ( ) ( )( ( ) )( )

1 2 1 2 1 1

3 1

11

2

2 22

k

k

k kk

k k

...

( )( ) ( )

11

2

k

k

k k

k

... 1

1

1

1

kk

...

1

1

0k

k

...

3

1

3

222 LiLi

= log log11

11 1

1

k kk

k

.46716002464644797643... 1 2 10

1

arcsinh arcsinh x dx

... ( ) ( )

!/2 2 1 1

11

1

2

2k k

k

k

ke

k

k

k

... 2

1

142 1

1

2

( )( )k

kk

kk

= ( )( ( )k k

kk

1 1

2 12

)

=( )!

( )!( )

k

k kk

1

2

121 2 1

x xdx x d2

0

2

2

0

1

cos arcsin/

x

1 .46740110027233965471... 2

214

12

2 1 1

( )!!

( )!! (

k

k k k )k

2 .46740110027233965471... 2

22

114

1

2

2

2 1

log( )

( )i

k k

kkkk

= 1arcsin2

= ( )( )

12

2

1

kk

k

k

k

!

=

0 sinh x

dxx

= log

( )( )( )

x

x xdx K k dk

1 10 0

1

3 .46740110027233965471... 2

141

2

2 1 1

( )!!

( )!! (

k

k k kk )

... E1

4

... 27 3

162 2

6

50

loglog

sin

x

xdx

... 2 11

1

2/k

k

.4678273201195659261... 11

22

1

kk

...

2 2 1

22

1 420

log

( ) sin( /

x dx

x x ) GR 3.552.9

...

2 2 1

2

1

1 40

log ( )

/

k

k k

...

1 2

)3(

2

722log8

kk

k

...

6

31

18

1

921

coth

kk

J124

... 1

32 3

3

110

Ik k

k

k

!( )!

= 15

32 5

2

1

kk

k

.46894666977688414590...

3

4

3

8

4

20

cossin x

xdx

.46921632103286066895... H kk k

k

kkk k

1

2 2

11

1 1( ( ) )

( )log

=

1

)12()2(

k k

kk

=

1 1

)( 1)1(1n k

kn kH

2 .469506314521047562476... 1

1 kk !

32 .46969701133414574548... 4

3

.469981161956636124706... 2

21

1 .47043116414999108828... 2

51

1

2 21

sinh

( )

kk

= 8

17

... 3

2

9

4

1

2 3

1

1

( )

( /

k

k k k )

...

3

23

5

4

3

322

coth

kk

= ( ) ( )

1 3 2 11

1

k k

k

k

.4709786279302689616... 1

31k

k k ( )

.471246045438685502741...

15

342

)2/1()5(

8

31)3(

2

21

82 k k

k

... ( )( )

( )log arctan

12

2 11

12 2

11

12

1

k

k k

k

k k kk

k

403 .4715872121057150966... 1

4 2

2 22

0

e ee k

ke e

k

sinh

!

...

( )

( )

3 1

5 1

4 .47213595499957939282... 20 2 5

...

Lik

k

kk

3

1

31

1

2

1

2

( )

...

0

2/3

22

3

4

3

2

3

2

5xx ee

x

=

0 13/2xe

dx

.47279971743743015582... 4 3

27

1

3 12 20

dx

x x( )

... 2

3

4

9 3

12 11

k

kk

... ( )( )

( )!sin

12

2 1

1 11

1 1

k

k k

k

k k

k

.47290683729585510379... 6 4 1 4 11

2 1 1

1

1

cos sin( )

( )! (

k

k k k k )

... si

x

dx

x

( )sin

1

2

12

1

... ( )

1

1

1

51

k

k k

... 3

22

1

2 11

2

0

e k

k

k

kkk

kk

! ( ) 2!

GR 1.212

...

4 6

3

2

1

5

1

4 421

tan

k kk 5

... 7

1440

7 4

16

1

2

4 1

41

( ) k

k k

= 9

19

771 .474249826667225190536...

( ) ( )4 1

2744 5

... 7 3

42 2

42

4 2

2 2

21

( )log log

( )

H

kk

k 1

... 2

12

... 2 2 4 3 1 32

( ) ( ) ( ) ( ) ( ) ( )k

k

k k

= k

k kk

3

5 42

1

.47480157230323829219... 26

3 3

1

6

2

1

log

kk k

k

k

...

8

3

3

4

1

4 1

2

20

hg hgx

xdx

sin

( sin )

... 1

1 k kk !

... 1

2 21k

k k

... ( )k

kk 41

... e

ek

kk

k

k

3

5 33 2

5 6 5 61

1

1 3

4 3

31 3

3

2

3

21

3

3

/

/

//

/ /

sin cos ( )( )!

1 .4752572671352997213... x dx

e xx

2

20

... Ai

1

2

... 71 5/

...

12

)2(log)(

k k

kk

...

( )( )

( )

( )

3

2

1

82 1 1

1

12

1

2

2 32

k kk k

kk k

.476222388197391301...

k

k

kk

k 2

)!1(

)1(4,2,

2

1F1

1

1

11

... 3

2 2

( )kk

k

... 3

21

...

1

3 2 33

1

4 121

cot

k kk

... 2

3 2 33

1

322

cot

kk

... erf

1 1

2

... log10 3 5 .47722557505166113457... 30

... 3

2

.47764546494388936239...

sinh

log

=

1

1 )2()1(11log

kk

k

k

kii

... 2

2

2log


... 8 6 2 7 22 1

22

1

log log( )(

k H

k kk

kk 2)

... 3 3

42

1

4

2

2

2

2

1

2

1

2

( )log

i i i i

= ( )

1 1

5 31

k

k k k

... G

G1

... 4 2

... ( )( )

( )! /

11

2

1

22 1

2

k

kk

k

k

k k

e

... 4

2 11

1

1

2 10

log ( )k

k k k

... 4 3 4 ( ) ( )

.47923495811102680807... 2 2 2

21

2 1

1

21

csc( )

/

k

k k 4

... sin( )

( )!

1

2

1

2 1 22 10

k

kk k

AS 4.3.65

=

4

1

16120

( )

!( )

k

kk k k

... Jk k

kk

k2

1

0

2 2 12

2

( )!( )!

.47962348400112945189... 2 2 11

21

cik k

k

k

( )( )

( )!

... ( )

( )!

/k

k

e

kk

k

k

1

1

2

1

1

...

3 3 2 41 3 30

/

dx

x

.480000000000000000000 = 12

25 4

2

1

Fkk

k

...

2 0 )(

1)(

k k

k

...

2 2

1

10

1

log log log

logx

dx

x

GR 4.229.3

... log( )

( )2

1 1

22 1

2 11

H

k

H

kk

kk

k k

k

= log( )( )

( )

x x

x

dx

x

1 4

2 20

GR 4.299.1

... 2 22 1

2

1

log( )

k H

kk

kk

1 .48049223762892856680... ( )2 1

2 12 11

kk

k

... ( )

! ( )

1

2 1

1

1

k

k k k Titchmarsh 14.32.3

2 .480548302158708391604... 22 e

1 .480645673830658464567... 2 2 1

11

2

2

2k

k

k

k

k

ke

( )

!/

1 .480716542675186062751... 2 3 1 12

213

2

k

k k

kk

k ( )

6 .48074069840786023097... 42

... 2 3

5

( )

... 1

8log

.481008115373192720896... 2 2 2 2 5 21

22

5

4arctan log log arctan log

=

112

1

0 )12(2

)1(

)22)(12(4

)1(

kk

k

kk

k

kkkk

= log( )

11

2 20

xdx

... 0

1 11

1( )

! (

k

k jk jk

)!k

... 2log2 log 24 -8 5 = arccsch

1

22

5

1

cos( / ) k

kk

... 2 2 2 4 31

2

2 21

( ) ( )log

( )

x

x xdx

... ek

k

kk

3 2

0

3

2/

!

... 615 1 3

0

1

ee dx /

x

... ( log )logsin tan

/1 2

22

0

2 x x dx

... 2 3 4 22 1

22

1

( ) log( )

H

k kk

k

... 24 1

2

2

0

loglog( )

cosh /

x

xdx GR 4.373.2

... 15

8

3

2

3

2 3

2

1

logk Hk

kk

7 .4833147735478827712... 56 2 14

... log

( )( )( )

3

2

3

21

k

k km

k m

... ( ) ( )

( )!sin

1 4 2

2 1

1

02

1

k

k k

k

k k

...

1

24

2

6

2 3

3

1

2

1

43

2 3

3Li Li( )log log

= 84 4

2

31

2

0

log x

x x

x dx

ex

...

2

1

41 2 1 2 ( ) (i i

... 4

14

1

2121

( )!

( )!

k

k kk

Dingle p. 70

1 .4844444656045474898... 3 18

2 3

3

3

3

2 2

30

log log log xdx

e x

2

... 3

64

... 2

1

3 1( )kk

... Ei

e

H

kk k

k

( )( )

!

11

1

=

log( )1

0

1 x

edxx

8 .4852813742385702928... 72 6 2

... 5

48 2

2

4

1

2

1

2

2 2

2 2

logLi

iLi

i

= log( )

( )

1

1

2

20

x

x x

...

1 )6(

11

k kk

...

6 3

3

6

1

2 3 13 10

log ( )

( )

k

kk k

Berndt 8.14.1

= dx

x x2 12

... ( )

1

1

1

61

k

k k

... 2 3

3

1

1

3

20

logx

xdx

= log( )

11

3 120

xdx

=

0 cos2

cosdx

x

x

... 4 210

Gx x

xdx

logcos

sin GR 3.791.2

= log( GR 4.224.10sin )10

x dx

... ( ( ) ( )4) 2 3 13

2 11

k

kk

k

9 .486832980505137996... 90

1 .48685779828884403099...

1

)1)()(k

kk

... 10 161

2 30

ek kk

k

! ( )

...

2 23 33 4 40

/

dx

x

... tan/

x dx0

4

... 12 2

1

1 220

csch

( )

/

k

k k

... 2

22 1

( )

( )

k

kk

.48808338775163574003...

15

364 4 1414 143 4

4 4

/ csc csch

= ( )

1

1442

k

k k

.4882207515238431339...

( )k kk

k 21

... 3

42 2

3 3

22

0

1

loglog

log( )x x dx

... 2 2 25

8

1

4 1

4

20

4

arctancos

sin

/

x

xdx

... 2

1 3/

... H k

k

k

kk

k k

1

2

2

2

1 1

2 1

( )log

.48883153138652018302... 2 4 2 2 23

42 log log

( )

= ( ) log ( )1 12

0

1

x x

xdx

... cosh ( )( )

2 21

2

8

22 2 2 2

0

!

e ek

k

k

... 3

5

... 3

22

22

2

1

2

1

log ( )! ( )

e

Eik k k

k

k

... 32 2

2 32 32 2 3

0

( )logx x

edxx

... eG

22 .49008835718767688354...

1

32

2)2()1(13 k

kkk

... ( )

12

1

41

k

k k

kk

... 13

16 12 12 1

2 2

1

log ( )

k

kk

k

Adamchick-Srivastava 2.24

...

2

1

2

40

9

5

2

( )

...

( )32

... 1

2 k kk ! ( )

... 5

32 12 40

dx

x( )

= sin5

30

x

xdx

GR 3.827.9

= dx

x( sin )5 20

2

= x GR 4.523.3x3

0

1

arcsin dx

...

( )

!log

k

k kEi

k kk k

2 1

1 1

k

1

= 5369

36005

6 H ( )

... e i i2 4

... 1

3

12

3

2 3

1

3 20ee

k

k

k

cos( )

( )

!

... dxe

xdx

x

xx

e x

0 022 1

)/sin(

1

cot

1

... 1

6 3 3 1 3

31 3 1 3 2 3 1 3

3

32 ( ) ( )/ / / /

k

kk

...

( ) ( )3 13

21 1

2

1

k

kk

k

...

41

1

2 1

2

0

4

cos

sin

/ x

xdx

... 16e

... 2

123

0

ex

xdx

sin( tan ) GR 3.881.2

...

4 2

2 21

16

1

821

coth

kk

... 2 2 1 2 2 2 1 2 21

2 2 30

arcsinh

log( )( )k

k k

2 .4929299919026930579... 23

2 622 2

0

x xd

ex

log2 x

... 5 3log

... SinIntegralk k

k

kk

1

2

1

2 1 2 2 12 10

( )

( )! ( )

... root of Ei x Ei x( ) ( ) 1

... 3

29 2

2

9

4

9 93

1 3

1 3 1 3 1 3

/

/ /cos cos cos /


.493480220054467930942...

2

22 1)2(

)2(2

)4()2(

20 nT

,

where n has an odd number of prime factors [Ramanujan] Berndt Ch. 5

.493550125985245944499... 1

2

22

2 k

k

kk

log

.4939394022668291491...

43

3

3 21 0

1

156 2

1

( ) log logx

x xdx

x

xdx

= x

e edxx x

3

0 1

6 .4939394022668291491...

43

156 4 1 ( ) ( ) AS 6.4.2

= log3

0

1

1

x

xdx

GR 4.262.2

= x

edxx

3

0 1

Andrews p. 89

.49452203055156817569...

( ) loglog ( ) log ( )

( )2 2 2

2

3

3

4

1

12

3 2

20

1

x

x x

.494795105503626129386... 2 3 12

2321

k

kk

kk

( )

... 51 4/

... log

log

cos 1 2

0

x

dxx

... G8

... 1

16

1

43

1

2 4

2

20

1

, ,log x

x

=i

Lii

Lii

2 2 23 3

.49566657469328260843... ( )4 2

4 41

2

41

k k

kk 1k k

=

8 2 2 2coth cot

... 1

2 11 ( )kk k

... 3

3 ( )

... 6G

... 2

125

4 53

31

sin /k

kk

GR 1.443.1

.496460978217119096806... 7

720

1

2

1

41 1 1

41 4 1 4 3 4 / / /csc ( ) csc ( )i

= ( )

1 1

8 41

k

k k k

... e

e

dx

x

x

1

4 1

22

20

sin ( )

3 .49651490659152822551... H

kk

k

22

1

...

16

2 2

2

2

21 1

cosIm

( )

( )

k

k

k

ikk

k

...

2 10 2 520

dx

x

... 31 6

32

7 4

8

2

2

1

2 4 2 2

( ) ( ) ( )coth tanh

= ( )

1 1

8 61

k

k k k

...

1

2

1

2

5

4

1

21 4

/

.497203709819647653589... sin

!

k

kk

kk 20

.497700302470745347474... 2 6 2 21

log log log ( )( )

log

k

k kk

Titchmarsh 1.1.9

=

primep p 21


6 .497848411497744790930... 7 3

4

3

82

2

16 6 1

2 2 1

2 2

2

2

31

( )log

( )

( )

k k k k

k kkk k

... Im ( ) i

... ( ) ( )

1 14

2

k

k

k

... 12 2 1

22

1

1eEi

H

kk

kk

k

( ) log ( )!

... sin sinh( )

( )

1

2

1

2

1

4 21

k

k k ! GR 1.413.1

.49865491180167551221... k

kk 2 330

... e ek

k

( )

2

... 1

2 22

kk k

( )

... eG

.49932101244307520621...

4

224

2

)1)(log

1dss

kk

k

k

.50000000000000000000 = 1

2

= 1

11 ( )!k k!k

=

1

)12()2(k

kk

= k k k

kkk k

( )

( )

2 1

4

4

4 112 2

1

= 1

3 30

2

1k

kk

k

k

= 1 1

2 122

312k k

k

k k k kk kk

( / )

1

= 1

4 121 kk

J397, K Ex. 109d

= 2 14 3

72

k k

k kk

= k k

kk

2

1

1

2

( )!

GR 0.142

= 1

1 3

1

2)0 1k k k k kk k!( )( ) !(

= 2 1

1 12 21

k

k kk

( )(( ) 1)

K 134

= 1 1

33 11

3 12k k k k k kk k

= ( )!!

( )!! ( )( )

2 1

2

4 1

2 1 2 2

2

1

k

k

k

k kk

J390

= 2

2 30

k

k k k k!( )( )

= ( ) ( ) ( ) ( )

1 1 2 22 1

k

k k

k k 1k

= ( ) ( )2

4

2 1

41 1

k k kk

kk

k

= ( )k

kk 2 11

= ( )2 2

4 11

k k

kk

= ( )sin

( )sin

( )sin

1 1 11

1

12

21

3

311

k

k

k k

kk

k

k

k

k

k

k

=

1

22

241)12(

13 kk

k

kFkk

k

=

122

1)2(exp

11

kk k

k

k

=

122

11

k kk GR 0.261

= 11

2 1

2

2 12

2

21

kk k

k

k GR 0.261

= k k

kk

( )

( )

2

1 21

J1061

= x dx

x

3

40

1

1

= log( )1

31

x

xdx

= log sin log log coslog2 2

11

x x dx x x dxee

= cos sin sin sinx

edx

x

edx

x x

e

x x

ex x x0 0 0

2

0

x

x

= e x x dx

sin cot0

=

0

11

4xe

dxx

= arctanx

xdx3

1

= dx

e ex x

20

= x e dxx3

0

2

1 .50000000000000000000 =

1

2

32

3

kkkF

= ( )sin

131

1

k

k

k

k

=

0

15

4xe

dxx

2 .50000000000000000000 =

1

2

)(2 2

)4(

)2(

2

5

k

k

k

HW Thm. 301, Titchmarsh 1.2.8

=

primep p

p2

2

1

1 [Ramanujan] Berndt Ch. 5

3 .500000000000000000000 =

21

2

32

7

kk

k

... ( )

arctan log1 2

8

1

42

2

16

5 2 2

5 2 2

2

81 2 1 2

3

8

2

16 2 62

arctan arctanlog dx

x x

.500822254752841076485...

( )( )

( ) ( )42

2 12

1

48

1

2 4 4 Li e Li ei i

= cosk

kk4

1

GR 1.443.6

.50112556304669359364... 3

9

1

6

5

3

1

12

7

3 2 42

arctan logdx

x x

.501268927971592850757... ( ) ( )

!!

1

2

k

k

k

k

.50132565492620010048...

0

6

222 cossin

5

2dx

x

xxx

... sin4 1

... 1

1 ( !)!kk

.50138916447355203054... cosh cos cos

( )

1 1

2 4

1

4

1

2

1

4 20 !

e

e kk

2 .501567433354975641473... SinhIntegralx

xdx( )

sinh2

0

2

.50197827418667468842... 1

9 31 k kk

.502014563324708494568... Likk

k7 7

1

1

2

1

2

.502507086021695538022... i i2

16 2

1

2

1

2

1

2

2 2

1 1

( ) ( )

4

2 2

16 2coth

= H

kk

k 4 221

.502676521478269286455... 1

261 k

kk

k

.502733573141073270868... 1 2 3 4 5 6 7 ( ) ( ) ( ) ( ) ( ) ( )

= 1

8 71 k kk

1 .503008799888373907939... H

k kk

k !

1

2 .50307111052515473419... 121

kk

kk

15 .50313834014991008774... 3 2

2 202

x dx

e ex x/ /

.503150623555038937007... 2

1

2 11 2

40

ex x

xdx

/ sinsin

Marsden p. 259

... 1

22 4 6

2

18 6

1

( ) ( ) ( ) cothk kk

...

0 )12(!!

1

k kk

... 1

8 51 k kk

... e e

... 1

2 90 2 2

2 2

2 2

14

8 41

sin sinh

cos cosh k kk

... 1

23

1k

k

... 1

8 31 k kk

... sinh 2

... 11

1211 2 2

5

2

1

4 21

HypPFQx

xdx{ , }, , ,

sin

... Likk

k6 6

1

1

2

1

2

... 11

e k

k

... 8

3

... 2 2 22

0

2

21

Ik

k

k

k

( )( !)

... 675 5 7

12 1

4 5

30

1 ( ) log

( )

x x

x

... ( ) ( ) ( ) ( ) ( )2 3 4 5 6 11

7 61

k kk

.5056902535487344733... sinh3

39001

814

4

kk

... 1

10 2 15

5

3

1

3 520

coth

kk

... ( ) ( )k kk

1 12

...

2 3

)(2)1(3log

331

kk

kk k

.506117668431800472727... 13

125 1 ( )

.506476876667430480956... log arctan3

4

2

2 4 2 22

dx

x x

... 5

2

... 2

... 3

32

1

1 30

log( )

/

k

k k

... 1

3 2 3 32 3 1

1

2

3

23 3 1

1

2

3

22 3 2 3 2 3 2 3 2 3

/ / / / /( )i

ii

i

1

3 21 22 3

1 3/

/

= 1

231 kk

.50746382959658060049... 1

7 41 k kk

... 12

3

1 142 4 4

4 41

csch csch

( ) ( )k i k ik

... logk

kk 22

... ( ) ( ) ( )/ /31

41 1 1 13 4 3 4

i

41 1 1 11 4 1 4 ( ) ( )/ /

= 1

7 31 k kk

.508074963143952841655... 5 2 2 3 ( ) ( )

.50811059680068603330... 1

7 21 k kk

... kH kkk

( )

1 11

...

1

0

1

0

2

1

)1log()3(

4

2logdydx

xy

xy

.508228442902056261461... 1

71 k kk

... ( ( ( )))2

.50835815998421686354... e

e kk3

2

3

3

2 3

1

3 20

cos( )

!

... Likk

k5 5

1

1

2

1

2

...

coth

2 2

4 226

11k

k kk

= ( ) ( ) ( )

1 2 2 21

1

k

k

k k

... (( )!)

( )

k

kk

1

3 1

3

1 !

... 3 22

1

4

1

421 1

log( )

k k

k

kk

k

= dxx

x

1

02

4 )1log(

...

3

2

1

2

1

2

7

4

3

23 4

/

=

3

2

3

4

1

2

1 4

/

... 3 5 4 2 31

41

( ) ( ) ( ) ( )

( )

k

kk

.50914721058334683065...

8 22

2

2 1

2

2 21

csch2

2 sinh( )

k

kk

=( ) ( )

1 2

2

1

1

k

kk

k k

2 .5091784786580567820... cosh cos(log ) sin ( ) 2 2

1

i i

= 11

2 1 20

( )kk

...

12

2

2

2 11

kk k

k

k

kLi

.510572548834680863663... HypPFQk kk

1113

22 2 2

1

4

1

2 21

, , , , , , ,( )!

... 2 4 21

12

340

log( )( )k kk

... 1

1 p kk ( )

...

21 11 2

2

0

2

J J x x( ) ( ) cos(cos )cos/

dx

... log5

3

2

5

2

511

Lik

k

kk

... 1

241 k

kk

k

... 3 2 2

20

2

20

3

8 3 3 1 3

log log logx

xdx

x

xdx

... 1 2 3 4 51

6 51

( ) ( ) ( ) ( )k kk

... 12 2

1

1 2

1

21

csch

( )

/

k

k k

... 1

3

2

3

3

2

1 3

33 3 2

5 6

0

1 3 1 3

e ek

k k

k

/ / /

/

cos( )

( )!

... kk

kk 31

... 1

3

2

3

3

2

3

33 3 2

5 6

0

1 3 1 3

e ek

k

k

/ / //

cos( )!

... 1

31

6 3 3

61

6 3 3

62 31 3

7 6 2 32 3

7 6 2 3

//

/ //

/ /

( ) ( )

i i

2

6

= ( )

( )

3 2

3

1

3 113 2

1

k

k kkk k

... ( )k

kk 2 11

.51301613656182775167... sin ( )

!( )

2 11

20

k

k k k

... ( ) ( )

1

2

1

1

k

kk

k

9 .513507698668731836292... 1

10

... 8

... sinh2

2 21

2

11

kk

... ( ) ( )3 1 3 1 11

k kk

... 285929

11664

1

7 77

7

1

Lik

kk

... 1

7 7loglog e

... ( )!

1 1

1

k

k

k

k

.51394167172304786872...

1 )13(3

32

kkk

k

Berndt Sec. 4.6

=

1

2

3

)(

kk

ku

...

2

4 21

2coth ( ) ( )

= 1

21 2 4 1

11

16 4

1

( ) ( )k

k k

kk k

... 4 4 20

tanhsin

sinh

x

xdx GR 3.981.1

.515121478609336050647... 4 16

2

2

2 2

31

log arctan x

xdx

... 3 3

7

( )

...

( )33

1 3

6

3 3

2

1 3

6

3 3

2

i i i i

= 1

21 3 3 1

11

16 3

1

( ) ( )k

k k

kk k

... log( )

( )

log

( )2

12

1

1 1

2

22

2

30

1

k

k

k

k

x

x

... 3 12

1

k

k

k ( )

... 81 5/

... 1

4 4

1

4

2

4

2

4

i i

i

= cos( / )x

ex

2

10

... F k

kk

3

1 8

... 2

6

2 2

2 2

sin sinh

=

23 4 1 4 1 4

6

1

41 1 1 1 1 ( ) ( ) ( )/ / /

1

41 1 1 1 11 4 3 4 3 4( ) ( ) ( )/ / /

= 1 1

21 4 26 2

1

1

1k kk

k

k

k

( ) ( ) 1

= Im( )

` 12 2

1 k k ik

...

... 1

2

22

2 1

0

2e ek

kee e

k

e

/ coshsinh

!sinhsinh GR 1.471.1

= coshcosh sinhcosh sinhsinh2 2 2 ... ( )3

... 5

64 162

2 2

32

2

20

loglog x

x xdx

...

0

2

xe

dxxx

.51684918394299926361... log

arctan7

6

1

3

5

3

3

6 132

x dx

x

... 1 1

21 5 16

1

1

1k kk

k

k

k

( ) ( ) 1

... 21

4 43

2

21

Lix

x xdx

log

...

2 2

)(

12

1

kk

primepp

k

...

1

1

...

6

1

2 6

1 3

2

1 3

2coth cot

i i

12

2 2 32

2 3i icot

= 1

1

1

21 66

1

1

1kk

k

k

k

( ) ( ) 1

.51716815675825854102... 8

2 3log

... 1

2

21

0

e ek

ke e

k

/ cosh( / )

! J711

= coshcosh sinhcosh coshsinh1

2

1

2

1

2

.5172859549331002461... 1

3 1 333

11 ( )

( )k k

kk k

k

... Likk

k4 4

1

1

2

1

2

... log( )11

e kk

k

... 6 2

2

CFG D1

... 11

31

k kk ( )

... 4 3 2 2 ( ) ( )

... sinh

( )

e

e

e

k

k

k

2 10 !

.51913971359001577609... 2 2

2

1 1 12

1

( ) ( )

!

k

k

k

k

!

...

12

)2(log)(

k k

kk

... 2

19

... ( ) ( )2

2 1

2

2 122

1 1

k kk

kk

k

... ( )2

2 11

kk

k

... H

kk

kk 31

... 1

2

1

1

Eie

Ei ek

k kk

( )cosh

!

... primep

p 22

115

Berndt 5.28

=

1

2

)(

)4(

)2(

k k

k

Titchmarsh 1.2.7

=

squarefreeq

q 2

.519837072045672480216...

1

122 1)2()1(1cschcoth4

1

k

k kk

=

222

2

)1(k k

k

1 .520346901066280805612... 2

10

1e

x

x xdx

e

( log )

/

GR 4.274

.520450870484383451512... 1

4 21 2 1 2 1

1

21 4 1 4

1 4i ii

( ) ( )/ /

/

1

4 21 1 23 4 4 ( ) / /

= k

kk4

1 2

.520459029022087280114... sin sin coshcos sinhcos sin sincos2 2 2 2 e 2

= sin

!

2

1

k

kk

.520499877813046537683... erfk k

k

kk

1

2

2 1

2 2 12 10

( )

! ( )

... log( sin )cos

2 12

1

k

kk

... ( )k

kk

2

2 2

= 25

48 4

1

4

1

4

1

44

21

21

23

H

k k k k kk k k

2 .520833333333843019783... 1

0 ( !)!!kk

...

log log

log(log )2

3

41

42 2 1

2

20

1

x

xdx GR 4.327.1

=

log

cosh

x

xdx

0

GR 4.371.1

.521095305493747361622... sinh( )!

/ /1

2 2

1

2 1 2

1 2 1 2

2 10

e e

k kk

AS 4.5.62

.521113369404676592224... 1

2 2 12

1k k

k ( )

Berndt Ch. 4

... arccot cot csc ( )sin

2 2 2 12

21

1

kk

k

k

k

.521302552157350760576... 2 1 21

21

CoshIntegralk kk

( )( )!

.521405433164720678331... 1

152 2 5 1 2 log( ) , avg. dist. bet. points in the unit square

.5216069591318861249... 1

2

1

2 0 0

e

x dx

e

x dx

ex x

sin cos

... ii

ii

eie

eie

22

22

1sin)1(

= 1)()(sin2

kkk

... ( )2

11

k

ekk

.521934090709712874795... 8

15

1

4 2 1 2

2

2 10

arcsin( !)

( )!

k

k kk

.522507202193083409915... ( )

( )log

k

k kkkk 2 1

1

21

1

212

k

... sinh2

101

42

2

kk

... 1

1442 kk

8 .522688139219475950514... 1

9

... arccot ( cos )cscsin

2 1 121

k

kkk

... 1

21 1 1 1

3

22 2

1

4

12

31

HypPFQk

kkk

, , , , , , ,

.523073127217685014886... 8 340

3

2 1

16 1 20

k

k k kkk ( )( )

... H k

kk

( )3

1 3

.52324814376454783652... 3 3 4 2 11

1 30

log log log( )( )

x xdx

... arctan

...

02 9

2arccsc)1()2(6 x

dx

= x dx

x

x dx

x

x dx

x1 1 1120

2

60

9

120

=

0 3xx ee

dx

... 12 2

1

1

22 0 1

0eI I

k

k

k

k

k

( ) ( )( )

( )!

... 3 3 4 ( ) ( )

... 21

2

1

2 1

3

20

4

arctancos

sin

/

x

xdx

... ( )

( )!!

1

1

2

1

k

k

k

k

... 1

20 ( )!k kk

!

... 2 2cosh

... log7!

... ( ) ( ) ( )4 3 2 11

5 41

k kk

= 1

21 41

1

( ) ( )k

k

k 1

... e

ek

k

k

k

3 2

5 3

5 6 5 63 2

1

1 3

4 3

33

3

2

3

2

3

3

/

//

/

/ //sin cos

( )!

1 .52569812813416969091...

3 2

3 2 1

12

3 1

2 11 2

//coth

( )( ) ( )

k k

k

k 1

... 2

5 5

1

2

3

5 csc

... 1

3

... ( )

( )

k

kk

11 !

...

( )

( ) ( )

3

3 4

= 10

19

... ( )

log

1

2

k

k k k

... H

kk

kk 2 4

1

... ( )

1

150

k

k k

... 5

16

27

5

5

20

logsin

x

xdx

...

1

1

!

)(

k k

k

...

arctan1

1 e

... 7

3

2 1

70

k

k kk

... 1

21 1 1 2 2 1 1

11

( )( cos ) log( cos ) sinsin

( )

k

k kk

...

2

42

7

12

1

21 2 2 12

2

1 1

1

coth ( ) ( )

k

kk

k k

k

=

12 32

1

k kk

.527887014709683857297...

4 2 21

1 2

1

1

log( )

( /

k

)k k k

=( )!

( )!( )

k

k kk

1

2

12

121 2

...

22

1

21log

1)2(22csc2log

kk

k

kk

k

... log8 3

... 1

3

6 3 3

6

6 3 3

61

1

3

7 6 2 3 7 6 2 3

1 3

/ / / /

/

i i

= ( )

( )

3 1

3

1

3 113

1

k

k kkk k

... 24

0

1

ee dxx

... 1

21

12

1

SinhIntegralx

dx

x( ) sinh

... G

... 1

2 1 20k

k

/

...

4 2 2 4

4140

2

3

2 2

34 3 2

2 1

4

log log( ) log

k

k kkk

=9

17

... 1

31 F kk

... k

kk

!

!

12

... 121

kk

k

... ( ) ( ) ( )31

2

1

22 2 i i

= 1 1

21 2 35 3

1

1

1k kk

k

k

k

( ) ( ) 1

... 1 1

22 k

k

kk

... 1

21 ( )!k kk

...

22

2

2

322

2 3 1 1 14 3

1

( ) ( ) ( )( )

k

k k

k kk k

k k

1

... 1

2 11 k kk ! ( )

Titchmarsh 14.32.3

... 8

... ( )( )

log ( )

1 21

1

k

k

k

kk

... ek

k ee

kk

1 1

1

/

!

... 3

2 2 1 3

3 4

40

/

/

dx

x

... 82

36

2 1

12 1 20

k

k k kkk ( )( )

...

2

3

2

20

1

12

1

2

1

3

3

83

1 3

Lix

x xdx( )

log

( ) ( )

... 41

22 1 2

1

1 4

1

1

log( )

/

k

k k

... 34

3

2

2

21

k kk ( )

... 2 1 2 20

4

log cos log(sin )/

x x

dx

... 7 1

3

3

1

3 2

1

e k

k

k

k

( )!

...

3

16

1

61 3

3 3

21 3

3 3

2

2

( ) ( )i

ii

i

= 1

21 3 2 1

11

15 2

1

( ) ( )k

k k

kk k

... ( )k

k

1

2

1

... 62 1623

4

1

ek

k kk

!( )

... log log ( )3 2

3

2

3

2 2

32

1

2

3 3

2

Li

=log

( )( )

2

1 1 2

x

x xdx

1 .53362629276374222515... e

.5338450130794810532... 2 1

2

( )

( )

k

kk

.53387676440805699500...

1 2

12sin

2

1

2

3sin2

2

1sin

51cos4

1

kk

k

1 .53388564147438066683... | ( )| k

kk 2 11

... 1

8

... 9

8

... 16

3

4

31

41

1

log

Hkk

k

.53464318757261872264... log x

xdx

2 121

... 1

2 44 1 4

0

e ek

ke e

kk

/ / cosh

!

.534799996739570370524... log 2 2

... logcos

sin

/

11

2 10

4

x

xdx

...

1

4

1

2

1

2

1

2

1

2

i i i i

= 1

21 4 1 1

11

15

1

( ) ( )k

k k

kk k

... sinsin

coshcos

sinhcos

sinsin(cos ) /1

2

1

2

1

2

1

21 2

e

= sin

!

k

k kk 21

GR 1.449.1

.5351307235543852976... 3

31

1

3 21

sin

kk

... 1

12 k kk !( ! )

... 1

21 F kk

... ( )31

3

... Ai( )1

... 2 2 3( ) CFG D1

... 1

21 5 1

1

11

15

1

( ) ( )k

k k

kk

.5360774649700956698... ( )2 1

2

1

2 12

0

dx

ex

...

e

... 1

21 6 1 1

11

16

1

( ) ( )k

k k

kk

k

.5365207790738756208... 1

3 1 322

11 ( )

( )k k

kk k

k

... 1

21 7 2 1

11

1

2

71

( ) ( )k

k k

kk

k

... 1

31 1

1 3

21 1

1 3

22 3 1 3( ( ) ) ( ( ) )/ /

i i

1

= ( ) ( ) ( )

1 3 1 31

1

k

k

k k

... ( )jkkj

2 111

... k

kk 4 11

... 1

2 2

2

21

sinh

Re ( ) i

... 7 3

8

2

6

2

12

1

2

1

2

3 2

3 31

( ) log log

Likk

k

=log( / ) log1 2

0

1

x x

xdx

...

0

3/1

)!3(2

3cos

3

2

3

1 2/)3/1(3/1

k

k

kee

... 5

2

1 5

2

1

2

1 2 2

2 1 2

1

2 11

log( ) ( )!!

( )!!

k

kk

k

k J141

... H ( )/

31 4

... sin!

1

1 kk

=7

13

... 2 2 2coslog i i2

.539114847327011158805... 1

8

1

4

1

44

1

2i

i itanh tan csc

i

1

84

1

2

1

4

1

4i

i ii

icsch cot coth

= e x

edx

x

x

cos

1 2

.53935260118837935667...

( / )

( / ) ( / )

( )1 2

5 4 3 41

1

21

k

k k J1028

... 1

31

4 2 3 6

21

4 2 3 6

22 31 3

1 3 5 6 1 32 3

1 3 5 6 1 3

//

/ / //

/ / /

( ) ( )

i i

2

32 6

1 3

2 31 3

/

//

= 6 3 16

613

2

k

k k

kk

k ( )

9 .5393920141694564915... 91

... e Not known to be transcendental

... 3 3log 1

... cos cosh Re( )

( )!1

1

20

i ek

ik

k

AS 4.3.66, LY 6.110

.540345545918092079459... 4 4

1

2cosh

.540553369576224517937... 2

21 28

21

2

4 1

2 1

1

2

log( )

( ) ( )k

k k

k k

k

k

kk

... sin(cos sin ) coshcos

sinhcos

1 12

2

2

2

= ( ) sin

!

1 2

2

1

1

k

kk

k

k

6 .540737725975564600708... 37

4 2

2

8

3

0

k

k

kk

k

... ( ( ))3 ... G7

8 .54097291002346216562... 3 2 3 31

3

30

1

( ) ( )log

( )

x

x

... 13 2 10 20 1

6

21

I Ik

kk

( ) ( )( !)

... 4 2

0

1

180

1

log( ) logx x

x

.5411961001461969844... 2 2

2 8 81

1

4 21

cos sin( ) k

k k J1029

.54123573432867053015... dx

x( )0

1

...

1 !

)1()1log(

k

kk

kk

Be [Ramanjuan] Berndt Ch. 5

... B

kk

k ( !)20

... 1

20k

k !

... log ( )( )3

2 6 3

4

81 3

1

3

2

3

26 3

27

31

= k k k k

k kkk k

2

2

2

313

36 9 1

3 3 1

( )

(( )

... 2 1

20

22

83

4

1

3 4

1

4

Gk

k k

kk

k

( )

( / )

( ) ( )

... 2

cosh

... 112 96 21 2 3

21

log( ( )(H k k kk

kk

)

... 1

2

1

2

1

2 1 21

sinh( )!

k k

k

... p

pp prime !

...

2 4 2 8 2 2

2 2

coth

i i i

i i

81

21

21 1 ( ) ( ) i

= H

kk

k 4 121

... 2

11

1 1sinh

)!2(

1

2

211cosh

x

dx

xk

ee

k

...

0

1

)!2(

1

2cos1cosh

k k

eei GR 0.245.5

= 14

2 12 20

( )kk

J1079

... 1

21 1 12log( ) log( )(sin cos ) e e e ii i i 1

= sin k

kk

11

...

1

)2(log)(

k k

kk

8 .5440037453175311679... 73

...

2tanh

2coth22

sinh

2

ii

...

1

)2(log

k k

k

... log

log sin( )/2

44

0

4

x dx

.54445873964813266061... ( ) ( )k kk

2 11

.5456413607650470421... dxe

eerf

ex

x

1

4/1

2

2

11

2

... 7

609 3

1 1

4 4

31

4

30

1

( )log

( )

log

( )

x

xdx

x x

xdx

1 ...

2

1

22 2 2 32

0

arctandx

x x

... ( ) ( )

1 112

kk

k

F k

1 ... F kkk

12

1 ( )

... 1 3 220

1

log log( )

/e

e

e edx

x

x

... 27 3

2 1

2

20

1

Gx dx

x

( ) arccos

... 2

118

1 1

2

( )!( )

( )!

k k

kk

!

= 2

1arcsin2 2

= dxx

x

0

3)1log(

11 .548739357257748378... i isin( log )2

.54886543055424064622... 2 1

1 2

1

2 23

2

1

4

1

2 1F

k

k

k

k

, , ,( ) ( !)

( )!

...

1 )12(2

1

kkk k

... log( )

( )

( )

( )2 1

2

0

01

1

2

1

1

Titchmarsh 2.12.8

... log

( )

3

2

1

2

1

2 2 12 10

arctanh kk k

AS 4.5.64, J941

= dx

x x

dx

x

dx

ex( )( )

1 3 102

2 0

2

=

12)2(

log

x

x

.549428148719873179229... 1

41 1 21 4 3 4csc ( ) csc ( ) cos( cosh(/ / ( 2

( ) sin ( ) ( ) sin ( )/ / / / )1 1 1 13 4 1 4 1 4 3 4

= ( )

1

140

k

k k

7 .5498344352707496972... 57

5060 .549875237639470468574... 8

157! 8 7

1

81

7

0

1

( ) ( )log( ) x

xdx GR 4.266.2

2 .550166236549955... ( )/

1 11

31

kk

k

e

k

... 3

20

... 6

G

... 3 6

... 6

6

1

2 12

( )

... 1

2

1

2

12

22 2

0

coth( )kk

...

1

2

2

)(

kk

k

... 4 2

2

1

1 12 2e e sinh sinh cosh

J132, J713

.5516151617923785687... ek

kk

13

6

2

1 ( !)!

.55188588687438602342...

0

1

13

13log

34

1

10

1

121012

)1(

k

k

k

.5523689696799021586... 8

1 11

4 31

erf erfix

dx

x

cosh

...

1 )5(

11

k kk

... sin

11

21 kk

... 11

5

1 21

21

( ) ( )!

( !)

k

k

k

k

... 2 2 2 11 2

2 10

sin cos( )

( )!( )

k k

k k k

2 .553310333104003190087... 0

2 1

( )

!

k

kk

... arctan 2

2 220

dx

x x 5

... 2 1

21

k

k kkk

!!

1 .55484419730133345731...

1

2)12(

21

2cosh

3

1

k k

...

4 2 820

dx

x

.5553968826533496289... 1

2

1 1

2 0

1

sin cos cos

e

x dx

ex

=coslog x

xdx

e

21

... 2

6

2

3

2

6

2 2

2 21

124

0

1

loglog logx

xdx

.55663264611852264179... 1

2 4 4Li e Li ei i( ) ( )

= ( )cos

1 14

1

k

k

k

k GR 1.443.6

... 21

2

1

2 1 1 21 21

log / /k

kk

...

0

2

)2(

)41(4)1(1tan

k

kkkk

k

B

6 .55743852430200065234... 43

.5579921445238905154... H k

kk

2

1 4

4 2

3

3

log log

... 1

6 6loglog e J153

.5582373008332086382...

132k

kk

k

H

... 1 2 2 22

2 10

cosh sinh( )!( )

k

k k k

... ( ) logk k

kk

1

21

... log

( )! ( )!

log

!

k

k n

k

kk n

n

k

1

1

12 1 2

... Jk

k

k0 221

2( )

( )

( !)

... 2222

3

1

ek

k

k

k

!

...

( )

( )

k

kk

1

12

... 1

21 k kk !!( )

...

1

1

11 4

3)1(

4

3

7

3

4

7log

kk

kk

kLiLi

... 2739 3

2

1

1 3

1

1 3

1

3

1

31

( ) ( )

( / )

( )

( / )

k k

k k k

= 14

251

21

2

1

2

0

( )sink k

kk

x

k F x x

edx

... 1

2 2 11 k kkk ! (

)

Titchmarsh 14.32.3

.56012607792794894497...

1 3

11

kk

=

12/)3(2/)3( 22

3

1

3

1)1(1

kkkkk

k Hall Thm. 4.1.3

... 9 121

3 11 3

0

ek k kk

k

/

! ( ( )

2

... 10

34 2 1

1

2

2

2 122 2

log ( )( )

( )k

kk k

k

k k

... 0

21

( )

( !)

k

kk

...

13

)2(3

)3/1(

127

3

1

2

127)3(13

33

2

k k

...

1

2

1

3

121

33

0

( )

( )kk

... 4 3 3 4 ( ) ( )

...

0 !

)1(

k

kk

ke

= k

k

ek

/1

1

11

KGP ex. 6.69

=n

nn loglog)(lim

HW Thm. 328

23 .5614740840256044961...

4 2 22 4

0

3

208 3

( )log xdx

ex

.56173205239479373631... 1

22 12

2

1

e ek

kk

cos cos(sin )cos

!

... 3 3

0

12

8

9 3

16

log ( ) arcsin

x

xdx

= 9

16

= k kk

k kk k

( )( )

2 1 12 1

12

2

2 22

... 1

22 2 1

1 1

1

log cos( ) cos

k

k

k

k

... ( ) ( )2 4

.56264005857240015056... 1

2 21

1

4 121

sin

kk

J1064

... ( ) ( )

1 21 2

1

k

k

k 1

... 2 23

2 23 2 2

22

1

log log( )kH k

kk

... pd k

kk

( )

21

.5634363430819095293... 6 21

4 300

ek k

k

k kkk !( ) !( )

= 1

1 30 ( )! !k kk

... 1

4

1

6

2

3

11 11

32

0

( ) ( ) ( )

( )

k

k k

... 1

2

1

2cosh

1 .56386096544105634313... 6 2 6 43

3

3

8

6

1

2

42

( ) ( )

( )

( )

k

kk

= ( ) ( ) ( )( ( ) )

1 1 12

k 1k

k k k k

.56418958354775628695... 1

2120

k

k kk ( )!

... 17

16

8 7

0

x

xdx

sinh GR 3.523.10

...

3

1

22

1

4 Berndt 8.6.1

... ( )( )

log

11

1

11

1

2

k

k

k

k k

k

.56469423393426890215... sin k

k

kk 21

... erfik k

k

k

22 2

2 1

2 1

0

!( )

... 11

31

31 1

kk

kk

Q k( )

...

2

2

0

tanhsin

sinh

x

xdx

... 61 4/

... ee

ee

ee

11

11

11cosh cosh int sinh int

1

2

12

1

2

12

1

2

12e

ee

ee

ecoshint sinhint log

= H

kk

k ( )!21

.565159103992485027208... I 1 1( )

... 1

2

1

2

1

2

42 2

1

Lii

Lii k

kk

cos /2

...

1

31

33

cos)1(

2

1

k

kii

k

keLieLi

... 2 24 1

2

20

4

arctancos

sin

/

x

xdx

...

cos cosh( )

( )!2 2

1 64

4

1

0

k k

k k

... 1 2 2 12

41

6

1

cos cosh ( )( )!

kk

k k GR 1.413.2

... e e eEi Eik H

kk

k

1 1 1 11

2

1

( ) ( ) ( )( )!

...

2

33

3

9

3

9

2

3

1

33

2 3

3Li Li( )log log

= 83 3

2

31

2

0

log x

x x

x dx

ex

... 2 2 2 12

11

21

Jk

kk

k

k

( )( !)

... 8

2

4

1

4 1 4 20

log

( )(k kk )

... Ik k

k0 2

0

21

2( )

( !)

... ( )k

kk 31

... 1

6

... 4

.5665644010044528364... e Ei Eix

edxx

21

0

1

2 1 21

( ( ) ( )) loglog( )

9 .5667536626468872669... sinh

2

21

22

1

kk

... k

kk

!

( !!)30

... log ( ) ( )( )15

8

3

21 1

1 3

22

k

k

k

k

k Srivastava

J. Math. Anal. & Applics. 134 (1988) 129-140

... arccot2

... 3 62

3

1

3 10

log( )(k

k k k 2)

... 1

21 ( )k kk

... ( )e

... log( )

log

k

k kkk k2

12 1

1

212 1 k

... 1

2

1

21

2

121

1

e k

kk

k

( )( )!

5 .56776436283002192211... 31

... 2 32

3

4 2

3

2 1

4

2 3

31

( )log log

k

k kkk

...

2

4 216 2

1

2

1

cothk kk

= 1

21 2 2 1

11

12

1

( ) ) Im( )

k

k k

kk k i

5 .568327996831707845285... 3 2/

.568389930078412320021... 1

2 2

1

22 12

( )kk

kk

jkj

... 3

41

3

2

2 3

0

1erf

ee dx

/

x

...

2 1

2 112

00

( )

( )!!( )

k

k kH

... ( )( )

12

1

1

kk

k

k

... e

... 1

31

6 3 3

61

6 3 3

64 31 3

7 6 2 32 3

7 6 2 3

//

/ //

/ /

( ) ( )

i i

1

31

1

34 3 1 3/ /

= ( )3

3

1

3 113

1

k

kkk k

.569451751594934318891... 4 2 5 3 ( ) ( )

... 1)2()(sinh1

kkk

.5699609930945328064...

( )

( ) ( )

log log2

2

1

2 122

2

k

k

p

pk p prime

Berndt Ch. 5

... Eik kk

k

1

22

1

21

log!

... H o

k

kk 31

... coshcos sinhcos cos sinsin sin (coshcos sinhcos cossin )1 1 1 1 1 1 1 1

= sin

( )

k

kk

11 !

2 .57056955072903255045...

1

2

2

)(

kk

k

.57079632679489661923... 2

12

1

sin k

kk

= 2 1

4 2 1

2 1

2 2 11 1

k

k k

k

k kkk

k

( )

( )!!

( )! ( ) J388

= ( )

/

1

2 1

1

21

k

2k k

= arccos

( )

x

xdx

1 20

1

= tanh x

edxx

0

= x dx

x x( ) sinh /1 220

GR 3.522.7

= x x

x

log( )1

1 20

dx

GR 4.292.2

1 .57079632679489661923... 2

i ilog

= ( )

/

1

1 20

k

k k

= k

kk

!

( )2 10 !!

K144

= 2

22

2 1

2 1

4 2 11

2

0 0

k

k

k

k kkk

kk

k

k

k

k k

( !)

( )! ( )

= ( )!!

( )!!

2

2 1 21

k

k k kk

= arctan( )

2

2 1 20 kk


= arctan( )

1

2 1 5 21 kk


= 4

2 1 2 1

2

1

k

k kk ( )( )

Wallis’s product

= dx

x

dx

x

dx

e ex x20

2 201 1 2

( ) 2

= dx

x x20 1 2

/

= sin sinx

xdx

x

xdx

0

2

20

= x

xx dx

2

2 20

1

1

( )log GR 4.234.4

= cos GR 3.631.20

/2

0

2

x dx

= arccosh x

x21

= log( GR 4.227.3 tan )/

e x d0

2

x

= 1

2 20 e ex x

= xx

dxlog 11

40

= ci GR 6.264 x x dx( ) log0

2 .570796326794896619231... 2

1221

k

k k

k

3 .570796326794896619231... 2

2220

k

k k

k

.571428571428571428 = 4

7

1 .57163412158236360955... 1

2 32 k kk

/2

.57171288872391284966...

0

1

13

13log

2

1

12212

)1(

k

k

k

.571791629712009706036...

1

2

1

3

1

3

2 1

31

( )kk

k

= 1

3 121 k kk ( )

3 .571815532109087142149...

2

3 1)12(128

111

4

)3(9

k

kk

.572303143311902634417... 4

91

4

3 41

logkHk

kk

.572364942924700087071... log1

2

=

e

e

dx

x

x

x

2

0 2

1

1 GR 3.427.5

...

2

112

1

42 2

k k kk

( ) ( )1

= 1

2 2 2Li e Li ei i( ) ( )

=

223 1k kkk

k

= ( )( ) ( )

11

21

2

k

k

kk k

= ( )cos

1 12

1

k

k

k

k

...

22

2

14

3

12 k

k

k

H

.57289839112388295085... 4 2

8

4 2

20

sin x

xdx

... 3 3

23 2

( )( ) ( )

... 11

21

( !)kk

.574137740053329817241... 2 2

216

1

2

cos k

kk

... 1 2

3

1

31 1

11 3 2 34

1

( ) ( )/ /

k kk

= 1

21 3 11

1

( ) ( )k

k

k 1

5 .57494152476088062397... e ee k

k

1 1

1

/ !

... 10 3

.57525421205443264457... x x

xdx

sin

cos

/

2 20

2

.57527071983288752368... 2

23

30

1

43 2

3 3

8

1

log

( ) log ( )x

xdx

... 16 12 31

2

0

1

( )

log x

xdx

... 12

1

2

1

21 40

e

erfik

k

k

k/

( ) !

( )!

1 .57570459714985838481... log

3

4

...

21 1

2

1

21 1

41 1 ( ) ( ) ( ) ( )( ) ( )i i

ii i

= ( ) ( ) ( )

1 22

k

k

k k k 1

... 3

30

4

321

2 1

2 1

( )cos( )

( )k

k

k

k Davis 3.37

...

21 1 21

1

coth ( ) ( )

k

k

k 1

= 1

14 2 42

2 1kk

k k

( ) ( k ) J124

= sin x

e ex x

10

... Jk k

k

k

k

k

k

k1

0

1

20

21

1

1( )

( )

!( )!

( )

( !)

... 11

41

k kk ( )

... (Euler's constant, not known to be irrational)

= ( ) ( )

log

1 11

1

2 1

k

k k

k

k k

k

= k

kk

k kk k

11 1

1 1

12 2

( ) log Euler (1769)

= 1

11

21 n

k

k k

n

kn !

log

( )

Shamos

= log log1

0

1

xdx Andrews, p. 87

= GR 4.381.3 log sinx x d0

2

x

= 1 2

0

2

sec cos(tan( )tan

/

x xdx

x

GR 3.716.12

= 1

22

1 12 20

x dx

x e x( )( ) Andrews, p. 87

= ( )/1 1

0

1

e edx

xx x Andrews, p. 88

= 1

1

1

0

e xe dxx

x GR 3.427.2

= 1

10

xe

dx

xx GR 3.435.3

= 1

10

xx

dx

xcos GR 3.783.2

= 1

1 20

xx

dx

xcos GR 3.783.2

... GR 6.264.1

10

si x x dx( ) log

= Ei GR 6.234 x x dx( ) log

0

.577350269189625764509... 3

3 6 tan

= 4

3

1

2 1 121 ( ) /kk

9

GR 1.421

= 2 1

6 10

k

k k kkk

( )( 2)

...

( )

( ) ( )

2

2 3

.577863674895460858955... 2 1 2

0e

x

xdx

cos

AS 4.3.146

= dxx

xx

021

sin

= GR 3.716.5 cos(tan )cos/

x x2

0

2

dx

= cos(tan )sin

xx

xdx

0

GR 3.881.4

= x x

xdx

3

2 21

sin

( )

Marsden p. 259

... 27 3

6 6

3

2

1

3

2

3 21

log

k kk

.578169737606879079622... ( )

( )

2

2

1

4

12

12 2

1

k

kLi

kk k

...

(sin sinh )

(cosh cos )

2 2

2 2 2 2

1

2

1

141

kk

= 1

21 4 1

11

12 2

1

( ) ( ) Im( )

k

k k

kk k i

... 3

2

( )

!

k

kk

... 4 log

= 11

19

.57911600484428752980...

2 4 2

12

412 24

7 3

( )( )

k

kk

1 .57913670417429737901... 4

25

2

=1

5 1

1

5 2

1

5 3

1

5 42 2 2 21 ( ) ( ) ( ) ( )k k k kk

6 .57925121201010099506... log !6

3 .579441541679835928252... 3

21 2 2

3

21

9

4 6

2

22

2

log ( ) kk kk k

... 1

12 k kk ( ! )

.5795933814359071842... 1

3 1 31

11 ( )

( )k k

kk k

k

... 2

32

2

1

2

20

2

e e

ex dx

x x

x( ) GR 3.424.5

... 2

3 1

22

20

log( )

xdx

x GR 4.261.5

.57975438341923839722...

3

22Li

4 .579827970886095075273... 5 G ... ( ( ( ( ))))2

... k

kk

k

k

k5

1 11

1

21 5 1

( ) ( )! 1

... e

k

k

kk

1

.58043769548440223673... arctan(tanh ) ( ) arctan4

11

2 11

0

k

k k


.580551791621945988898... ( )( )

13 1

1

1

kk

k

k

... 2 21

0

e k

k

/ !

... 3

43

1

3 13

2

0

1

( )log

( )( )

Lix

x xdx

3

...

036/13 33

2

x

dx

... k

kk

k

k

k

2

61

1

11

1

21 6 2

( ) ( ) 1

.581824016936632859971... 29

64 4

1 4 3

1

e k

k kk

/

!

... k

kk

k

k

k

3

71

1

11

1

21 7 3

( ) ( ) 1

...

12/1

1 /112

11

1

1

kk

kk k

eee


=

1 !

)1(

k

kk

k

B

= dxex

x

12)1(

log

... e

e e

B

kkk

kk

k

1

1 1

0 0

( )

!

.5822405264650125059...

122

22

2

1

2

1

2

2log

12 kk k

Li

J362

= ( )

1 1

1

k k

k

H

k J116

= log( )

( )

log1

10

1

21

2

x

x xdx

x

x xdx GR 4.291.12

= logx

x

dx

x

12

=

log( )/ 1

0

1 2 x

xdx GR 4.291.3

=

log 1

20

1 x dx

x GR 4.291.4

=

log

1

2 10

1 x dx

x GR 4.291.5

4 .58257569495584000659... 21

.583121808061637560277... 2G

... sin log1 2

= 7

12

.58349565395417428579... ( )( )

12 1

1 1

1

kk

k

k

10 .583584148395975340846... 3 1

2

2

1

2 2

0

e k

k

k

k

( )!

.5838531634528576130... 2 1 2 12

22 1

2

1

cos ( )( )

!

kk

k k GR 1.412.2

... 101 5/

1 .58496250072115618145... log2 3

.5852181544310789058... 16 8 212

0

1

loglog( ) logx x

xdx

...

2)(2

1

kk

... ( )

1

130

k

k k

... 2

1 2 10

4

dx

xsin

/

... 7

3

4

3

4

31

3 4

0

1

,

/

e dx x

... 0

1 2

( )

( )!

k

kk

...

8

9

... 21

... 1

3

2

3 32 1

22

1

1

arccsch ( )kk

k k

k

...

0 )!2()!1(!

11},3,2{{},

k kkkHypPFQ

... G

... 512

63

5

1 2

/

... 8

tanh2

...

0

32/

)!3(2

3cos

3

2

3

1

k

k

kee

... log( ) log( )

log

k

k

x

xdxk

k

1

2

2

1 0

1

GR 422.3

...

12

2

)1(

)13(3

16

3

)(

363

2

9

1

2

3log

kkk kk

kkk

...

7

8

.58760059682190072844...

132

1

1cosh

)!2(2

1sinh

x

dx

xk

k

k

...

3 2 25 6 60

/

dx

x

... sin5

= 17

10

... 9 3 32 1

6

2

0

k

k

kk

k

... i i i ik i k ik

( ) ( )( ) ( )( ) ( )

1 12 2

1

1 11 1

... ( ) ( )

1 21

1

kk

k

F k 1

... 3

2

3

2 3

1

1 3

1

21

csch

( )

/

k

k k

...

02 2)!2(!

122

kkkk

I

... 3 2

208 1

e

x

exdx

log

... 3

16

23

21

21

k k

k kk

( )

( )( ) J1061

=

032

02

6

0

5

)1(

sinsin

x

dxdx

x

xdx

x

x GR 3.827.12

.5891600374288587219... 8 2 2 4 21

23 21

( ) log( )

/

k

k k k

6 .58921553992655506549...

6

7

... e erfk

k

k

12

12

0

( )

!!

.589255650988789603667... 4/

0

3cos26

5

dxx

... log2

2 2

kk

k

1 .58957531255118599032... log

1

4

... e

xe

xdx

21 2

0

log log

.589892743258550871445...

)62(2

2

6324)31(

6

1 3/13/16/53/1

3/5 i

i

2

6324

6

)31( 3/16/53/1

3/5

ii

=

23 6

1

k k

.59000116354468792989...

12

2

)!2(

),2(

k kk

kkS

... ( ) ( )

arctan

1 1

2 3

1 1

23

2

k

k k

k

k k

k

.59023206296500030164...

0 )!2(

)1(

k

kk

k

B

... 243

518

2 2 1

60

k

k

k

kk

( )

3 .59048052361289410146... 2

9

.590574872831670752346... 6G

1 .590636854637329063382...

112

01 )!2(

2

)!()!1(!

1)2(

kkk k

k

k

k

k

k

kkI LY 6.113

...

5

1

4 .59117429878527614807... 2 2 1 4 2

0

2

arcsinh x dx

... log log( )1 2

21

12

0

1

2

xdx

x GR 4.295.38

= ( s GR 4.226.2in )/

1 2

0

2

x dx

...

1

3

!5

k k

ke J161

... 1

5 22 2

12

0

1

log sin( log )li

xx dx GR 6.213.2

... 5

...

5

6

9 .5916630466254390832... 92 ... 3log6

.59172614725621914047... dx

e ex 2

0

x 2 erndt Ch. 28, Eq. 21.4

1 .59178884564103357816... 21

2

1

2120

e erfk k

k

( )!

10 .5919532755215206278...

1

22

)!2(1cosh

2sinh2

k

k

k

11 .5919532755215206278...

0

2

)!2(2coscosh

k

k

k

eei

AS 4.5.63

=

1

2)12(

41

k k J1079

...

6

743234)3(182 )2(3

...

6

134)3(182 )2(3

...

01

4/1 sinh)!2(

!

2

1

2

2

dxxek

kerfe x

k

.592394075923426897354... 12

2

( )

( )

k

kk

...

1 2

sin

)4/11cos1(2

1sin

1cos45

1sin2

kk

k GR 1.447.1

... 2

)223log(

2

2log

22

22log

2

2

2

2log

=

1

12

2kkkH

...

12 3/1

1

322

3

k k

... log

( )( )

2

21 1 1

x

x xdx

...

12

)3(

2kk

k

k

H

...

1

3

2kkkH

... 231 e

... ii ee

i

1sin2

1

... csch

2 1

4 2112

1

2

1( )k

k k k

...

1 2

)1(13log2log1

kk

k

k

...

2

35

6

33

2

35

6

331

iiii

=

1

)13()13(k

kk

...

0 2)!2(

164

kkk

e

...

1 2

)1(44

kk

kpfe

...

1

0

1!

)(

k k

k

...

4

5

...

1

)2(log)(k

kk

... 2

4

... )!12(

33)1(

4

11sin

12

1

13

k

k

k

k GR 1.412.3

.595886193680811429201...

22 )1(

)3(23k

k

kk

H

.596111293068951500191...

H 1(

1

3 .59627499972915819809... log

...

e Eik

kk

( )( )

!1

1

0

=dx

e x

x dx

e xx x( ) ( )

1 10

2

0

=dx

x xe xx

21 01

1( log )

log( )

dx

... 4 11

4

0

e Eix dx

e xx( )( )

.5963475204797203166... ( )

1

2 1

1

1

k

kk

k

...

13)1(

log

4

1

2

2logdx

x

xx

1 .59688720677545620285... 11

22

1

kk

...

11

1)1(

kk

k

k

.59713559316352851048...

043 )1(3243

80

x

dx

...

1

2

)!2(

2

4

5

k

k

k

e

...

0

2

)3(!

2

4

1

k

k

kk

e

...

1 !

)22()2(

k k

kk

...

22/1

2

111)(

kk kkLikk

...

1

44 4cos2log

2

1)2sin2(log

k

ii

k

kee

...

0 )12(!

2)1(2

22

1

k

kk

kkeri

...

0

4

!

4

k

k

ke

1 .598313166927325339078...

2

324

1)2(128

127

48120 k

kk

...

1

2

4

3,1

224

kk

kk

G

.599002264993457570659... 2

1 1 1 1 2 1 11

12 2

2

1

cos sin sin log( sin ) sinsin ( )

( )

k

k kk

...

1 !

1

k kFk

... 1)3(log

... 3

)3(1

... log ( ) 21

kk

.5996203229953586595... 2 11

1 11

2

1

1

e Eik k

k

k kk k

( )( )! ( )!( )

=1

1 22 ( )!k k!k

= H

k k k kk

k k!( ) ( )!

2

1

11 1

1 .599771198411726589334... 2

11

k

kk k( )

...

1

24

4

)4(17

360

17

k

k

k

H

Borwein & Borwein, Proc. AMS 123, 4 (1995) 1191-1198

... arccsch2

= 3

5 42 1

1

F kk

k

.600053813264123766513... 3

2

5

24

2

22

1

22

1

2

2 2

2 2

logLi

iLi

i

= log( )

( )

1

1

2

2 20

x

x xdx

.600281176062325408286... 3 6 2 91

1 3

1

1

log( )

( /

k

k k k )

.6004824307923120424...

1

)22()2(

k k

kk

... e k

k

1

1

1/ !

.601028451579797142699... ( )

sinh

log

sinh(log )

log ( )

sinh(log( ))

3

2

1

1

2 2

0

1

0

2

1

0

x dx

e x

x dx

x

x dx

xx

.60178128264873772913...

22

12

1 111)1()1(

kk

k

kLi

kk

k

... 1

1 12 k

k

kk

log

... K1 1( )

... ( )33

5

... log10 4

8 .6023252670426267717... 74

2 .60243495809669391311... k

k kk k

2

21

21

2

11

1

1

...

2 5

51

10

1

521

coth

kk

... 24 5 50 32

3

5

21 1

4 24

2

) ( ) ( )

k kk

= 16 11 5 1

1

4 3 2

52

k k k k

k kk

( )

... 22

sinh

Berndt 7.2.5

1 .60343114675605016067... 2 2 21

23 arctan log

= log( )

12

1 20

xdx

2 .603611904599514233302... 11

1

k k

k

... log

!

k

kk

2

.60393978817538095427... 1

2

1

2 20

2

ee xx

/

/

cos dx

...

1

1

12

)(

kk

k

.604599788078072616865...

3 3

1

3 2

1

3 11

k kk

=

3 3

2

2 1 41

( )!!

( )!!

k

k kkk

= 1

3 1 3 20 k kk

)( )

= 1

21 k

kk

k

CFG F17

=dx

x x

dx

x x

x dx

x x2 2 2 4 2 120

20

4 20

1

= dx

x x

dx

x x20

202 4 3

3

=

03

06

3

06

123 811 x

dxx

x

dxx

x

dxx

xxx

dx

= dx

x x x x x5 4 3 20 1

= dx

e ex x

10

... log !8

.604898643421630370247...

1

1)1(

2

1,

2

1

2

121

k

k

k

... 10

... 7

48

3

83

9 3

16

1

4

2

3

2 21

log

log ( )

.605065933151773563528... 9

4 2

1

1

2 2

0

1

x x

xx dxlog

.605133652503344581744...

0 )1()1()1(

2

12

xe

dxEierfi

e x

... 9

8

3 2

4 2

2

22

log

( )

k

kkk 1

... sik k

k k

k

( )( )

( )!( )2

1 2

2 1 2 1

2 1

0

AS 5.2.14

.605509265700126543353... 3 2 4 4 ( ) ( )

... 112

2 2k kk

k log( )

... 13

... 5

256

3 43

31

sin( / )k

kk

GR 1.443.5

...

1

2)12(

21

2cos

k k

... log2

0

1

11

x

dx

... 4

11 2 2

0

ee xx cos dx

... dydxxy

yx

1

0

1

0

22

1

)log()3(3

... ( )( )

!/

12 1

11 1

2

2k k

k

k

ke

... 1 1

2

1

20 0ei

k ki

k

kk

k

k

/ ( )

!

( )

( )!!

... 5

.60669515241529176378... 1

2 1

1

2 2 1

2

211 1 2

kk

k kk

k

kk

( )

( )

1 .60669515241529176378... 1

2 1 2

2 1

2 2 11

0

1 12k

kk

k

k

k kk

k

( )

( )

=1

21

2

211 2jk

kj

k

kk

( )

Shown irrational by Erdös in J. Indian Math, Soc. (N.S.) 12 (1948) 63-66

... log 21

2 10

xe

edx

x

x GR 3.452.3

... 1

1 k kk ! ( )

.60715770584139372912... erf

e k

k

k

( ) ( )

( )

1 11

20

!

3 .6071833342396650534... 3 35

1024

15

1

( )

( )

( )

a kk


... 1536 5634

4

0

ek

k kk !( )

... 12 6 32 2 1

6 10

k

k kkk ( )

... 6 1

21

12 2

12

( )

( )k

k pk p prime

... 4

6081075

1

1

12

2 4 6 8 10 12

p p p p p pp prime

.60798640550036075618... 1

2

1

22

3

2

1

4 2 22 1 2 11

Fk

kkk

, , ,( )

1

=dx

x x2 22 2

... ( ) ( )2 5

... 3

2

3

2 31

log

Hkk

k

... 1

1 2 21 ( )! (k kkk

) Titchmarsh 14.32.1

... 2

31185

1

1

8

2 4 6

p p p pp prime8

.608699218043767368353... log logsinh

( )( )e e k

kk

k

4 2

12 11

1

K Ex. 124f

= log( ) ( )

log1

2 21

12

2

i i kk

... !

1)2(2

1 k

k

k

k

k

... Hk

kk

3

1 4

... log54

51

Li

... 2 2

3

1

3 40

4

sin

cos

/ x

xdx

... cosh( )

e e ee

ke e

k

k

1

2 2

2

0 !

...

2 4 3

302 15

9 31

( )( )

x

edxx

... log!

11

2

kk

.6106437294514793434... 2

3

G

... 8 2 e

= 11

18 3

1

3

1

33

21

24

H

k k k kk k

.611643938224066294941...

1

2 3

1

61 3

1 3

21 3

1 3

2

( ) ( )i

ii

i

= k

kk

k

k

k3

2

1

111 3 1

( ) ( ) 1

5 .6120463970207165486...

14

243 1

4 3

30

log x dx

x

1 .612273774471087972438... sinh

0

21

12

1i kk

...

12/3

1

2

14

2

3

k k

... 142

3 453 3 5

1

1

1 12 4

2 52

( ) ( )

( )k k kk

... 11

32

( )kk

... sinh2

21

42

1

kk

... 1

2 4

1

21

2 1

2 11

1

log tan ( )sin( )

k

k

k

k GR 1.442.3

... log6 3

3 .613387333659139458760... ( )

!

/k

k

e

kk

k

k

2

0

1

21

2 .613514090166737576533...

( )3

.613705638880109381166... 2 2 23

2

1

2

log hg

= 1

2 21 k kk

GR 0.234.8

= k

kkk 2 10 ( )

= ( )( )

( )

11

2

2 1

21 11

1 1

kk

k

k

kk

kk

= log x

xdx

1 20

1

= dxx )1log(1

0

2

= 4 1

1

1

2 1 220

1

0

1 log( )

( )

log( )

( / / )

x

xdx

x

xdx2

x

GR 4.291.14

= GR 4.384.10 log(sin ) sin/

20

2

x x d

4 .61370563888010938117... 6 2 22 1

3

1

log( )

k

kkk

128 .61370563888010938117... 130 2 22 1

5

1

log( )

k

kkk

.613835953026141526250... 11

11

2

1

3

1

321

tanh

k kk

.613955913179956659743... 2

5 5 5 2

7 10

50

/

dx

x

.61397358864975783997... 1

22 2log

... 4 42 1

1

1log ( ) /

k k

k

...

1 32

31

)1()!(

)!(

k kk

k

.614787613714727541536... 11

3 11

k kk ( )

... erfik kk

k

1

2

2 1

2 2 12 10

! ( )

=8

13

... arctancos

sin

/1

2 1 20

4

x dx

x

...

log(cos )( ) ( )

( )!1

1 2 2 1

2

1 2 1 22

1

k k kk

k

B

k k AS 4.3.72

7 .6157731058639082857... 58

...

1

2

1

2 2

1

22

1

2

1

22

1

2

21

22

1

2

cot

sin sin cos cos sinh sin

cos cos cosh sin

Adamchik-Srivastava 2.28 ( ) sin2 11

k kk

... arctan

!

k

kk

1

... 8 3 ( ) ... 2 4 4 4cosh e e

... 2 22 1

1

1 2

1

1log ( ) / / 2

k kk

k

k

k

...

1

2

2

10

2

)14(

4

4

)2(

1

)1(

16 kkk

k

kk

k

kkk

k

H

= x x

xdx

arctan

1 40

GR 4.531.8

... 2 3 6 2 31

12

130

log( )

( )( )

k

k k k

...

6

3

2

1

3

1

3 220

coth/

kk

... sincos

22

2

1

2

... 510

5 5

2 1 1

5 10

log( )

k

k kkk

... F

kk

kk 2 2

1

... 8

... k

kk

k

2

2 11 1

( )

...

3

7

2

3log3

324

15

...

0 05/12 316

5xx ee

dx

x

dx

...

11 5

2

1

... 5 1

2, the Golden ratio

... 1 2

... 1

21k

kk kH

... 4 2 22

20

log( )

k

kk

...

2 4

)(3)1(

4

2log9

8

31

kk

kk k

... 2

315

1

1

4

2 4

p pp prime

... 11

1 21

( )( )k kk

... 16

3

10 2

3

2 1

8 20

k

k kkk ( )

... 2 1

1

k

k k kk

... 16 6

1

3 121

cot

/kk 2

.620088807189567689156...

1

22

12

)2/1(

)1(2

2csch2

6 k

k

kk

... log( ) (

!

e k

k

k )

k

1

2

1 2 1

1

=

1 !

)12()1(

k

kkk

kk


= dx

e x10

1

... 4

9 3

1 3 2

1

e k

k kk

/

!

4 .620324229254256923750... 0

1 1

( )

( )

k

kk

!

.62047113489033260164... ( )( )

arctan

12 1

2 1

1 11

21

k

kk

k

k k

k

... 11

51

k kk ( )

... 1

31

4 2 3 6

22 2 62 3

1 31 3 5 6 1 3

2 3 1 3/

// / /

/ /( )

i

( ) /

/

/ / /1

3

4 2 3 6

2

2 3

1 3

1 3 5 6 1 3

i

= 6 3 1 16

613

2

k

k k

kk

k ( )

... 1

2

1

20, ,

... 2 2e /

... 2 2

2 224 22

2

2

1

2

1

2

log

logLi

iLi

i

= log( )

( )

1

1

2

20

1

x

x x

... 1

5 5loglog e

2 .621408383075861505699... 1

45 2 13 4 5 50 12 5 2 65 20 5

= 5 5 5 5 5 5 5 ... [Ramanujan] Berndt Ch. 22

... )1(22

1cos1sin)1(1cos)1(1sin)1(1cos

2sicicici

= sin arctanx dx

x

x dx

ex10 0

= dx

e xx 20 1

.62182053074983197934... ( )

( )

1

2 12

k

kk k

... 2 3

6

CFG G1

... sinh sinh sinh32 4

311

43 3 1

1 3 3

8

e e e

e

6

= ( )( ( )

!

3 3 1 1

81

k k

k k

) J883

= 1

4

3 3

2 1

2 1

1

k

k k

( )!

.62322524014023051339...

02 1422

1arcsinh

xx

dx

... 1

21 2

1

21

1

2

1

2log arcsinh arctanh

= 11

4 1

1

4 11

( ) ( )k k

k k k

= 1

2 2 11k

k k( )

GR 1.513.1

= ( ) ( )!!

( )!!

1 2

2 10

k

k

k

k J129

= x dx

x x( )1 12 20

1

=

02 142 xx

dx

= sin

cos

/ x

xdx

2 20

2

= dx

x22 2

... 1

2 330 kk

... 21 3

2log

... pr k

kk

( )

2 11

= 13078

1251

21

6

1

( )k kk

k

k F

.624063589774511570688... 1

6 4 3

3

2

1

4 320

coth

kk

...

24 5 50 32

3

5

21 1

3 24

2

( ) ( ) ( ) ( )k

k

k k 1

= ( ) ( ) ( ) (4 2 4 11

24 1 4

1 1

k k kk k

)k

= 16 11 5 1

1

4 3 2

52

k k k k

k kk

( )

1 .624838898635177482811... K 2 1( ) ... ( )3

= 5

8

.62519689192003630759... 2

3 2

8 3

27 2

2

20

2

sin

( sin )

/ x

xdx

...

1 3

)2(

3csc

3log

kk k

k

1 .62535489744912819254... 11

22 21

kk

... 1

4

AS 6.1.10

... 2 1

12

2

2

2

k

k

k

k

k

ke

k

( )

!

...

2 2 1

2

0

sech

e

e

x

xdx

/ cos

cosh GR 3.981.3

=

dxee

xxx

cos

.626059428206128022755... ( )( )

12 1 11

21

22

k

k k

k

kLi

k

... 22

3 2 21

16 21 31

( )log

( )/

x

x xdx

.626503677833347983808... 6 1

5 4 1 221

sin

( cos )

sin

k kk

k

.626534929111853784164...

( ) ( )2 21

k kk

1

... S k k

k

k

kk

22

1

2 2

2

( , )

( )

... 71 4/

... 1

2 2

... sinh( )! (

22

2

2 1

4

2

2 2 2 1

0 0

e e

k

k

k

k

k

k

k )!

dx

AS 4.5.62

= e GR 3.915.1 xx2

0

cos sin

.6269531324505805969...

( )

( )

3

2 1

1

213 3

1

kk

k k

k

.627027495541456760072... 3 3 4 2 31

3 221

log log/

k kk

.627591004863777296758... ( ) )2 6

... 2

3

1

3

2

3

1

3

2

3

321

,k

k k

kk

= dx

x x

x

xdx2

3

301

1

log( )

= dx

x10

2

cos

= e

edx

x

x

/3

1

= log log( )

1 14

23

0 0

x dxx x

dx

.627716860485152532642... ( )( )

log

13 1 1

111

1 23

k

k k

k

kk

k

... 1

42 1

1

2 10

e erfik kk

!( )

.6278364236143983844... 4

5

4 5

25

1

2 arcsinh

= 2 10

111

2

1

4

12

Fk

k

k

k

, , ,( )

... sin( )cos log( sin ) sin

1

22 1 4 1 12

= sin

( )

2

1 1

k

k kk

... 5 1

3 2

50

x dx

x

/

... sin ( ) sin/

1

2

3

21 3 i

= cosh( )

3

21

3

2 1 20

kk

1 .628473712901584447056... 1

11 k k

k

... Lik

k

k3 3

1

( )

... ( ) cot ( ) cot ( )/ / / 14

1 11

21 4 3 4 1 4

i

= ( )( ) ( )

// /

1

42 1 2 1

1 43 4 3 4

( )

( ) ( )/

/ /1

42 1 2 1

3 41 4 1 4

= ( ) ( ) Im( )

1 4 2 11

21

1

2

42 1

k

k kk

k

kk

k

k

i

14 .628784140560320753162... 2 35

64 1 1

9 11

1

22

2

2

32

( ) ( ) ( )( )

k kk k

k kk k

... ( )/

1 11

41

kk

k

e

k

... 0

1 112

1

22 2

( )kk

kk j

kj

2

... 24 21 164 2

0

dx

x( / )

... 2 1 2 112

1

e eEik k

kk

( )( )

!

1 .6293779129886049518...

2 12

22

2

2 2

20

loglog log xdx

e x

2

= 2 2

2 2

012

2

12

( log )

loge xx dx

1 .629629629629629629629 = 44

27 4

3

1

kk

k

... G

... 1

22

12

log( )

( )

k

k kk

= ( ) log2 2 11

11

21

kk kk

... 11

2

p

p prime

.6304246388313639332... 11

21

362 4

( )

... ( )34

7

... 2

e

... log3 2

... 3 1

4

3

3

3

0

sin sin

k

kk

Berndt ch. 31

... 1

2 11k

k k ( )

=12

19

... 105

16

9

2

... e dxxsin

0

1

.6319660112501051518... 9

4

1

3 21

F kk

GKP p. 302

.631966197838167906662...

1

2

2

22log

12)3(

kk

k

k

H Berndt 9.12.1

.632120558828557678405... 11 1

0

1

31

ex x dx

x

dx

xcosh cosh

.632120558828557678405... 11 1 1

11

21 0

1

e k xx e x d

k e

x( )

!( ) log x GR 4.351.1

=

13

1

0

1coshcosh

x

dx

xdxxx

... 22

.6328153188403884421... ( )

logk

k kkk k

1

3

11

1

31 1 k

1096 .633158428458599263720... e7

6 .63324958071079969823... 44

... cos( )e

... 3

1 3

... 1

11

0

( )k k

k

2 .634415941277498655611...

loglog sin1

2 1

2 2

20

e x

xdx

... Ei ee

k k

k

k

( )!

11

...

1

2

1

4/1

)1(

2csch2

k

k

k

... arcsec15 33

6

Associated with a sailing curve. Mel1 p.153

...

2

1

2

2 2212

13

161 2 1

2 1

1

( ) ( )( )

k kk

kk k

.635181422730739085012... )()1(22

2log

1

kk

k

=

xx

dx e x dxxlog log log1

0

1

2

0

GR 4.325.8

... 2

12

142

1

4

4

4 1

Gk k

kkk k

( )

( )

...

2 )(

)2(1

k k

k

.63575432737726430215... 2 2 2 31

42

1 32

( ) ( )( )

k

kk

= ( ) ( )( ( ) )

1 12

k 1k

k k k

... I k

k kk

1

21

2

2 2

( !)

... 1

2 2

1

120

csch

( )k

k k

= 7

11 63 1

1

F kk

k

... 2 1

1 222

0

logsinx x dx

... 2

3112 16

1

96

2

sin /k

kk

GR 1.443.5

= iLi e Li ei i

2 32

32 / /

... ( ) ( )2 7

... 2

11

4 21

kk

GR 1.431

= cos

2 11

kk

GR 1.439.1

= 0

1 2/

= i e ilog

... 1

5

1

51

1

43 5 2

5 5

24 5

( ) / i

1

51

1

43 5 2

5 5

21

1

43 5 2 5 51 5 2 5( ) ( ) (/ / i i

( )(

/1

5

1

43 5 2 5 5

1 5

i

= ( ) ( )

1 5 3 11

1

1

3

52

k

k k

kk

k

...

2 2 2

1

2

1

2 11

2

221

1

1

coth ( )( )

k

k

k

kk

k

... 6 1

4 2 5 2

2 2

1

sin

cos

sin

kk

k

... 3

2

2

12 2

e

x

xdx

cos

( )2

... 1

2 422

4

0

cosh( )

k

k k 2 !

... ( ) ( )k k

kk

1 1

2

1 .6382270745053706475... 1

2 dkk

... 26

442

21

cos k

kk

GR 1.443.3

.6387045287798183656... 3 9 3

24

1

6 421

log

k kk

= dxx

x

1

05

6 )1log(

.638961276313634801150... sin log Im2 2 i

... 1

3

5

3

5

3 1

4

0

3

x dx

ex

... i

i i

i i

k

kk

k2

21

22

1

2

21

22

1

2

14 2

2 11

1

log ( )( )

1

= arctan1

22 kk

.639446070022506040443... ( ) ( ) ( )3 4 2

.6397766840608015768... k

kk 3 11

... 3

8 83 3 43

43i

Li e Li ei

Li e Li ei i i( ) ( ( ) ( ) i

= sin3

41

k

kk

... 4

32 log

1 .640186152773388023916... 7

32 log

.64031785796091597380... dxx

xx

1

0

2

1

log

27

502)3(16

... H

kk

kk

( )

( )

3

1 2 2 1

... 1

2 2 2 1 2

21 3 1 3 2 3 1 3

3

32 ( ) ( )/ / / /

k

kk

... 22

e

... H ( )/

31 3

.641274915080932047772...

2 6 2 3 320

20

dx

x

dx

x 2

... 21

3 33

2

0

1

Lix

xdx

log

... 3

24

5

5

6

1

6

hg hg

1 .641632560655153866294... 1

2

1 1 2

1 1 21 2

2

2 111

k k

k

kkk

( ) /

/

/

[Gauss] I Berndt 303

... 10

... ( )

1

2

1

1

k

kk k

... ( )( )

( )!sin

14 2 1

2 1

11

12

2

k

k k

k

k k

.642092615934330703006...

122 1

1211cot

k k GKP eq. 6.88

=

0

2

)!2(

4)1(

k

kk

k

k

B

... cot1

... 7

G

.64276126883997887911...

4

32Li

.6428571428571428571 = 9

14

1 .64322179613925445451... 11

32 6

4 2 3 6

21 3

1 3 5 6 1 3

( )/

/ / /i

1

3

4 2 3 6

2

1 3 5 6 1 3

i / / /

= 6 3 1 16

613

2

k

k k

kk k

( )( )

.643318387340724531847... ( )

/

k

kLi

k kk k

1 1

21 2

2

1

.643501108793284386803... 2 22

2arctan log

gd

9 .6436507609929549958... 93

... e

x xesinlog sin log

1 1

2 1

dx

... e ee

kk

k

k

sin ( )( )

12 1

1

1 !

... Gi

Lii

Lii

log2

8 2

1

2

1

22 2

= GR 4.227.11 log( tan )/

10

4

x dx

=

log( )1

1 20

1 x

xdx GR 4.291.10

= arctan

( )

x

x xdx

10

1

GR 4.531.3

.643796887509858981341...

( )1 e

eHe

.643907357064919657769... 1

6 4 2

7

2

1

4 421

tan

k kk 6

...

0

2

3 2

sin

2log42log

1x

x

... 2 22

( )k

k

... log jj

kjk

2 112

... G5

... 11

41

k kk ( )

.6449340668482264365...

2

61 2 ( )

...

2

2)1(

2 1)2()2(1

6 k k

J272, J348

=

12

12 )1(

1

1

1

kk kkkk

= ( ) ( ) ( ) ( )k k k kk k

1 2 22 1

2

2 1 = ( ) ( ) ( ) ( )

1 12 1

k

k k

k k k k k

= 1

21

21k

dx

xk

=

x x

xdx

log

10

1

GR 4.231.3

= log x

x xdx3 2

1

= 1

1 1 20

1

x

x x

xdx

log

( ) GR 4.236.2

= x dx

e ex x

10

GR 3.411.9

...

2

12

12

162 1

1 2 1

1

( ) ( )/

( )( )

k

k

kk k

= H

k kk

k ( )

11

= 31

221 k

kk

k

CFG F17, K156

= k k k k kk k

( ) ( ) ( )

1 1 11 2

=

log x

xdx

10

1

GR 4.231.2

=

0 )1(

)1log(

xx

x

= log

( )

log

( )

log

( )

log

( )

2

20

1 2

21

2

30 11 1 1

x

xdx

x

x

x

xdx

x

x xdx

1

= log( )1 2 2

0

1

x x

xdx GR 4.296.1

= GR 4.223.2

log 10

e dxx

=

0 1dx

e

xx

=dx

e x

10

...

2

261 1

k kk

( )

...

2

262 1

k k kk

( ) ( ) 2

.64527561023483500704... 3

62 2

3 3

2 log

log

= log( )( )

11

1 20

x xdx

.64572471826697712891...

1

)(

k kF

k

... 7

... 3

31

1

31

sinh

kk

.645903590137755193632...

0

2

1

sin)21()21(

4

1

2 xe

xii

.64593247422930033624...

2

)1(cot

2

)1(coth2)1(coth25

16

1 iii

=

8

1

16

1

81 2 1 1 2 13 4 1 4 3 4 1 4coth ( ) ( ) ( ) ( )/ / / /

1

81 2 1 1 2 11 4 3 4 1 4 3 4( ) ( ) ( ) ( )/ / / /

=

1262

1)68(1

kk

kkk

... 11

2 2

.646458473588902984359...

1

33

18

3 4

3

2 2

3

1

3

1

42

2 2

2Li Li( )log log log

= x

edxx

30

.646596291662721938667... 11

25

24

25 5

1

21

1

21

1

arcsinh ( )!( )!

( )!k

k

k k

k

... 1

45 1 3 1 1

2 2

2

1

sin cos ( )( )

k

k

k

k !

... ( )

log( )6 4 1

11

4 6

2

k

kk k

k k

.64704901239611528732... 1

2

1 3

22 11

1

1k k

k 1

k

k

kk

( ) ( )

= 11

17

3 .647562611124159771980... 36 6

22

( )

...

0

2cosh)1(4

2

dxxee x

... 1

2

1

2

1

21

1

2 21

coshcos

sinhcos

sinsin sin

!

k k

k kk

.647793574696319037017...

dxx

x

1

3

arcsinh

2

21log

2

1

2

1

... cos3

2

... 3 3

2

22

1

log log( )

x

xdx

... sech12 1

212

0

e e i

E

kk

kcos ( )! AS 4.5.66

.648231038121442835697... 3

5

2

5

5

20

2

0

arctan log log cosx x

edxx

... ii eLieLi 22

2

2

1

3

1 .648721270700128146849... ek k

ikk k

i

1

2

1

20 0! ( )!!/

...

2

15

2

15log

4

3

12 22

2

Berndt Ch. 9

.64887655168558007488...

1 )24(

)4(

k k

k

... 4

1

2

3tanh

32

= k

kk

k k

4

62 11

6 4 1

( )

...

0 )2(3

13

2

3log9

kk k

... ( )

log( )5 3 1

11

3 5

2

k

kk k

k k

... 4/

04cos6

1

3

2log

3

x

dxx

... 2

1sin

= 20

13

1 .650015797211168549834...

1 !)1(

1log

k

kk

ke

kk

Be

e

e Berndt 5.8.5

...

1

2

)6(!15894320

k kk

ke

.650174578577121012503...

1

2

)6(!15894320

k kk

ke

...

2

3 23

41

3

4 31 1

log ( )( )k

k

k

kk k

...

3/1

3/23/2

3/1

3/13/1

3/2

2

)1(1log)1(

2

)1(1log)1(

3

2

3/1

3/2

2

11log

3

2

=

0 )23(2

1

3

2,1,

2

1

3

1

kk k

.6503861069291288806... iLi e Li e Li e

k

ki i i k

k212 2 2

12

1

( ) ( ) Im ( ) ( )sin

1 .650425758797542876025...

0 )12(!

121

k kkerfi

.650604594983248538006... 1

21kH

kk

...

1

3 )12(2)12(27

)1(412

2 k

k

kk

=

2/

02

22/

02

2

sin1

cos

cos1

sin

dxx

xdx

x

x

= 1

0

22

2

)arctan(1

dxxx

x GR 4.538.2

...

22

1

1arctanh

)12(

1)24(sinhlog

2

1

kk kk

k

2 .6513166081688198157...

1

2

3

428 3

12 33

43

0

( ) ( )( )kk

.651473539678535823016... 15

32

5

4

35

64 5

2

1

log

k Hkk

k

... 2log

.651585210821386425593...

2

2 2

2 21

4 2

29 4

1 4 1 4

1 4 1 41

1/

/ /

/ /

sin sinh

cos cosh( )

( )

kk

k

k

= 1

2 2 21 k kk

...

22

2 )(

k k

k

... 31

12601

6

0

1 log( )

logx

x

xdx

... )7()2(

...

1

34

34

3

2 4sin

23

16

3

24

k

ii

k

keLieLi

i

GR 1.443.5

...

2

2 1)2(k

k

...

1

3

!!2

1

211711

k k

kerf

ee

...

2

55

24

59

2

552514

20

1 ii

2

55

24

59

2

55251

20

1 ii

2

55

24

59

2

55251

20

1 ii

2

55

24

59

2

55251

20

1 ii

=

1232

1)35(1

kk

kkk

.653743338243228533787... ( )( )!

15 3

2

1

212 3

1

kk

k k

k

k k

... ( )

log( )4 2 1

11

2 4

2

k

kk k

k k

1 .654028242523005382213... coshcos sinhcos sin( sin )sin

!1 1 1 1

1

k k

kk

... 2 37

41 1 1

2 3 3 1

11

3

2

2 32

( ) ( ) ( ) ( )( )

( )

k

k k

k k kk k

k k

... /1e

... ( )3 1 1 1

21

2 22

k

kk Li

kk k

... 3

7

1

3

2

0

( )k

kk

k

k

... 11

61

k kk ( )

... )3(

8

... 163 1 1

42

1

Gk

kk

kk

( )( )( ) Adamchik (27)

.655831600867491587281...

1

1 !)1(

kk

k

k

k

... ( )2

2 1

c Patterson Ex. A.4.2

... ( )2 1 1

31

3 22

k

kLi

kk k

...

5

33Li

...

2

/12

2

2

)1(1

!

1)(

k

k

k

kekkk

kk

...

1

22

1

2

1

2

1coth

k k J951

... 4 2

3

2 3

31

42 1

1

log log

H kk

k

... 5log6

... dxx

x

1

06

2

1

)1(log

32

32log

2

32log34

1 .65685424949238019521...

0 8

112424

kkk

k

5 .65685424949238019521... 32 4 2

... ( ) logk k

kk 22

...

1

21

2

i i

...

2

3

2

3

22592

5533

4 iLi

iLi

i

=

13

6/sin

k k

k GR 1.443.5

... )3()4()2(

)4(

15

2

=

primep kk

k

primepk

k

pp

p

k 022

2

12

)( )1(

1

)1(

HW Thm. 299

=

12

)(

k k

k HW Thm. 300

1 .6583741831710831673...

22

22

062 2 2

2

log loglog ( )x dx

ex

.65847232569963413649... 2 2

20

12

4

7 3

8 1

log ( ) arccos

x x

xdx

= dxx

x1

0

2arcsin

... 45

4)4(8

4

... dxx

xex x

2

2/

0

tan

cos

4sin

2

3 2

GR 3.963.3

.658759815490980624984...

24

14

111)1(

kk kLi

kk

k

.658951075727201947430... 1

21

1

21

Eik kk

( )!

... 4

3172

31

( )( )k

kk

... )1sinh(sinh)1cosh(cosh1)2sinh2(cosh eeee

=

02 )2(!

1)1(

k

ke

kk

e

e

ee

...

2 1

/1 11

!

)()1(

k k

kk

ke

k

k

...

k

k

k

k

8

)1(2232

3

240

8 .6602540378443864676... 75 5 3

.660349361999116217163... 11

3 121

k kk

...

1

2

2

22

2 )2/(sin

4

1

128

1

4 k

ii

k

keLieLi

... 4

21

8

1

421

coth

kk

J124

=

0 1

cossinxe

dxxx

...

044/9 22 x

dx

...

05

2

15

3csc

555

2

5

2dx

x

x

... G

...

22 1

log

k kk

k

...

1 !

)1(

k

kk

k

kB

...

0 !

2cosh2sincosh2coshsinh2coshcosh

k k

k

... log cos log ( )( )/ /21

4

1

21 12 2

e ei i

= ( )cos /

121

1

k

k

k

k

... 7)32log(42)21log(21362174105

1

avg. distance between points in the unit cube

... 1

4

3 3

8

23

1

log log( )x

xdx

... )6()2(

...

1 )!(

1

k k

...

1

3

k

k

k

...

4

1

8 14 2 1

2

42 1

coth ( )

k

kk

k k

...

1 3

1

k kF

...

1333

4/cos

3

1

3

1

2

1

k k

kiLi

iLi

... )3(

2

... 41

12

120

Gk k

k

k

( )

( )( )

...

13

41

)2(3

)(

164

4

1

2

164)3(28

k k

...

03

41

)2(3

)(

1

4

1

2

1)3(28

k k

=

1 )1(4

12

140

93

kk

kk

4 .66453259190400037192...

0

2)12(

21

2cosh

k k

... 0 1 0 20

11 2 1

Fe

Je k e

k

kk

; ;( )

( !)

.66467019408956851024... 3

84

0

2

x e dxx

= dxxx

xe x

2/

06

2tan

cotcos

cos212

GR 3.964.3

.66488023368106598117...

2

061 1 2

1

( ) ( )logx x

e edx

x x

... ( )

log( )3 1 1

11

3

2

k

kk k

k k

... 2 2

1 .66535148216552671854...

5

212

55

55log55log5

10

3

5

6

=

2 5

1)(6

kk

k k

...

1 2k

k

kk

kH

... 28

...

arctanh1

1 e

.665738986368222001365...

12

2 )(

k k

k

... 4

arctan

... 25

...

20

12

222

)!(

1)2(

kk kkI

kk

k

...

0 22

)21arctan(2

24

3xx ee

dx

...

23

3 )(

k k

k

...

12

/2

0 !

)2(2

k

k

k

k

k

e

k

k

... 22 ... 1sincos ... sin ( ) cos ( ) cos1 1 1 1si ci 1

= log sin( )x x10

1

dx

.666666666666666666666 =

k

k

2

)1(

3

2

=

1

1

0 )2/1)(2/3(

)1(

)2)(1(4

12

k

k

kk kkkkk

k

=

1

34

8kkkF

=

1

2)2(

11

k k

=

23 1

21

k k J1034, Mardsen p. 358

=

2/

02sin1

1

dxx

= dxe

xx

02

cosh

= cosh log xdx

x31

=

0

8

3xe

dxx

1 .666666666666666666666 = 3

5

=

1

14

8kkkF

2 .666666666666666666666 = 3

8

=

1

4

8kkkF

...

0 )!!!(

1

k k

.666822076192281325682... 21

...

0 )!12(

3

32

3sinh

2

3cosh

k

k

k

k

.66724523794284173732... 21 2

2

1 1 12

1

log( ) ( )!

!

k

k

k

k k

... 4

arccos

...

2

1

42 22 2 e ei i

( ) sinkk

1 11

2 k

...

0 12

)1(

4

3,

2

1

4

1,

2

1

2

1

k

k

... ( )2 1 1

21

2 22

k

kLi

kk k

...

2

1

6

2log

6

2log)2( 3

32

3 LiLi

=

0

2

13

21

0

2

22

log8

21

logxe

dxx

xx

dxx

x

dxx

...

122

)(

kko

k

k

...

1 )1(12

1

kk k

...

0

323

1)3(122log82log2

xe

dxx

1 .6701907046196043386... k

kk 2 11

...

1

1

kke

... 55

... 3

2 3

...

0 )2(

111log1

kk kee

ee

.671204652601083... ( )

/

12 3

2

k

k k k 2

...

15

/11)1(

k

kk

k

e

1 .67158702286436895171... 2

12sin

!

11cossinh1coscosh1sin

2

1sin

0

k

kk

... e erf e

xdx

x ( )1 1

2 1

2

20

...

1

021

1loglog2log

6

5

6

1log

3

2

xx

dx

x

GR 4.325.6

=

01 1

log

ee

xx

.67177754230896957429...

033 )1(327

10

x

dx

...

13

1)()(

2

1

k kkii

=

1 1

1 )14()14(2

11)12()1(

2

1

k k

k kkk

=

12 )(

1Re

k ikk

.672148922218710419133...

1 )2/5(

1

5

2log4

75

92

k kk

...

12 24

1

2cot

244

1

k k

...

0

13

13

)13()1(

3

31log

3

1

34 k

kk

k

...

2

3 1)(2)(k

kk

...

2

3 )()(k

kk

...

2

3 1)(k

k

... 2

2cos

4

1

4)1cosh()1cos(

2

2cosh2cos2

2

e

eii

=

0 )!4(

16

k

k

k

.6731729316883003802... 1

2

3

4 83 4

2 4

20

/ cossin

x

xdx

... 1sinh

21

ee

.67344457931634517503...

1 )(

1

k kq

... 1

2 2

1

2

1

2

1

21

2

21

HypPFQx

xdx, , ,

sin

.6736020436407296815... kk

k

2

1 4 1

... 2

15log

8

3

20

2

Berndt 9.8

...

kLi

kk

k

kk

11

)1(

1)(

13

23

...

1

3

!8

))1(1)(33(1cosh

k

kk

k J844

...

1

31

22

)12(

)12(2

2

)1(6

42

)3(7

kkk k

kkk

3 .674643966011328778996... pk

kk 21

the kth prime pk

1 .674707331811177969904...

2 )1)((2

11

kk k

...

1 2

1)2(

k kF

k

...

22)1(

log

k k

k

... 1

pp prime !

...

0 )6/1)(1(

)1()32log(32log

5

6

k

k

kk

...

0

2 2

sin

2log1

1x

x

.67551085885603996302...

02 12322

2

1arctan2

2

2arctan

xx

dx

= dx

x x20 2 3

14 .67591306671469341644...

1 1 1

3

222

1

02

)21(

42

22

)(

n m mm

kk

mnk

k

knmkk

.675947298512314616619...

2/1

04

4/

04 1

1

sin1

cos

24

5log2arctan

22

1dx

xdx

x

x

...

0

2

)!12(2

0sinh

k

k

k

ee

i

=

12

11

k k

...

21

1arctan

1

12

1)2(

kk kkk

k

... ( )

!

k

kk

1

... ( )k

kkk 21

...

10

8/1

!)!2(

!)!1(

2!!

1

22

1

21

kkk k

k

kerfe

... ( ) ( )( )k kk

k

2

.677365284338832383751...

0 )2()!2(

11sinh141cosh1812

16212

k kkee

... 3

2 12

1

21

2

2

( ) ( )k k

k

... )()()(3)(38

33

3333

iiii eLieLieLieLii

=

13

3sin

k k

k

3 .678581614644620922501...

2

2

)!1(

)(

k k

k

17 .67887848183809818913... e erfk

k k

k

k

2

1

2 22 2

2 1

( )!!

( )!! !

... 1

3

... 2

1cot1

... 4

11

4 22

12 1 2

2

20

eK

x dx

x/ ( )sin

= GR 3.716.9 2/

0

2 )(tansin

dxx

...

1

02

6

77

7

1

log)()(360

256

61dx

x

xiLiiLii

GR 4.265

... 1

2 112j

kjk

...

2

22 1)(k

k

...

03

01

2

)2(!4

1

)!2(4 xe

dx

kk

kex

kk

...

2

1)(

k

k

k

kH

0 =

1

22

425

292

kk

kFk

...

22

1)(1)1(

)()1(

kk

k

k

kHkk

k

... 1

log1

1)(22

k

k

k

kkH

kkk

7 .6811457478686081758... 59

.68117691028158186735... 1

2 2

1

121

cot

kk

...

12

1

x

dx

... 4/18

... 3

2111 1

1

411 2 1

2 12 1 2 1

2

22

F ii

F ik

kkk

, , , , , ,( )

... 4/

0

2232

cot192164

2log dxxxG

...

1 11

0

1 3

1

3

)(

13

1

j kkj

kk

kk

k

=

1 )13(3

132

kkk

k

Berndt 6.1.4

...

1 !

11

k k

5 .68219697698347550546... dxe

xdx

xx

xdx

x

xx

1

3

12

31

02

34

1

log

1

log

4

)4(21

120

7

GR 4.262.1

.682245131127166286554... 11

22 2 2 2 2 2 1 1

1

k

k

k( )

... )2/(sinh2

... 3log5

... 11

71

k kk ( )

... 3/1

...

1

2 1)1(k

k

...

0

2)12(2

11

22cosh

k k

... )(1sin2 ii eei

... coshcosh sinhcosh sinhsinh1

2

1

2

1

2

J712

= 1

2

21

0

e ek

ke e

k

/ sinh( / )

!

.683150338229591228672...

12

)1(22

33

2

2

13log

8

93log

4

3

24

7

k

k

kk

H

...

1

1sin1log

1

k kk

...

1

0

2

1

1

!

22log)2( dx

x

e

kkEi

x

k

k

.683899300784123200209... ( )k

kk 21

... 2

22

12 2

162 2

2 1

2

2

4

log( )!!

( )!

( )!

( !)

k

k k

k

k kkk

k

=H

k kk

k ( )( )

1 2 11

=log ( )2

20

1 1

x

xdx

= 19

13

.6842280752259962142...

( )

( )

k

kk

1

22

21 .68464289811076878215... dxx

x

1

04

442

)1(

log)3(9

90

7

3

... log log( )

( )Kk

k jj

j

k

k0

1

1

2 1

1

22 1

11

K Khintchine's constant 0

= log ( )22 2

2

21

2

1

21

4

Li Lik

k

k2

= ( ) log21

22

1

12

2 22

Likk

=

kkk

11log

11log

2

Bailey, Borwein, Crandall, Math. Comp. 66, 217 (1997) 417-431

.68497723153155817271...

112 1

)(

)1( kk

kk

kk

... 11

2

2

k k

k

( )

log

Khintchine's constant

1 .685750354812596042871...

4

1K

...

2

11)(k

kk

...

2

3531

2

35312

6

1 ii

ii

=

1

1

13

1)3()1(2

1

1

1

k

k

k

kk

=

1212

1)13(1

kk

kkk

.686827337720053882161...

12 2

1

2

1

2

7tanh

7 k kk

.686889658948339591657... 7

)3(4

.6869741956329142613... 3

4

G

...

22 3

1)(

k k

k

... )4()2(3

... 8

32

3

2

3

tanh tanhsin

x

e edxx x

... ( )

cos( )/

2 2

29 42

0

2

e xx dx

.6885375371203397155... 11

41

kk

...

02/3

2/3

)12(

1

2

321

k k

...

02 )!2(!

1)2(

k kkI LY 6.114

...

2

/1

1

1!

1)2( 2

k

k

k

ek

k

3 .689357624731389088942... (log )log( )

2 25

1

2

20

arccschx

xdx

...

0

22

3

12log22log2

6 x

x

e

ex

...

2 5

)(4)1(

55

55log

5

15log

5

21

5

21

kk

kk k

... 2

arccsc

.69019422352157148739...

2 1 40

2

ee xx

/ cos

dx

7 .690286020676767839767... 3log7 4 .69041575982342955457... 22

... cosh

sinh cosh2

0

11

2

1

2 x x dx

...

0 )2(6

12

3

43

kk kk

k

... 1

22 2 3

3

1

log coscos

k

kk

... 1

21 1

1

2 1

1

1

(cos sin )( )

( )

k

k

k

k !

...

02

12 2/12

1

2/1

1

2cot

21

kk kkk

=

112

)2(

kk

k

...

211

2

0

2

2

0

2

J x x dx x x dx( ) cos(sin )cos cos(sin ) sin/ /

= sin(sin ) sin/

x x d0

2

x

...

1

4

481

380

kk

k

... 12 102

1

31

I

e

k

k

k

k

( ) ( ) ( )!

( !)

2

.691849430120897679...

sin( ) sinh( )

2 2

61

42 4

2 k

.691893583207303682497...

1

2

)2(22log42log24

kk

k

k

kH

... ( )

( )

2 1

21

k

kk

.692200627555346353865...

0

/1

!

)1(

kk

ke

eke

= 13

92

.6926058146742493275... 1

2 22 k kk log

...

1

1)1()1(2log

k

k

k J71

=

2

1

2

11

1

Likk

k J117

=

0

12 )12(3

12

3

1arctanh2

22

1arcsinh2

kk k

K148

=

2 2

)(

kk

k

=( )

log( )2 1

11

2

2

k

kk

k k

K Ex. 124(d)

=

112

112 42

)2()14(

2

)12()12(

kk

k

kk k

kkk

=

0 )22)(12(

1

k kk GR 8.373.1

=

11 1 )2(!)!2(

!)!12(

!)!2(

)!1(

)!2(

)!22(

kk k kk

k

k

k

k

k J628, K Ex. 108b

=

13 28

1

2

1

k kk [Ramanjuan] Berndt Ch. 2

=

1

3 28

)1(21

k

k

kk [Ramanujan] Berndt Ch. 2

=

1

3 327

)1(

2

3

4

3

k

k

kk [Ramanujan] Berndt Ch. 2

=

12 14k

k

k

H

=

1211

H nk

kkn

( )

=

1

cos

k k

k GR 1.448.2

= limn k

n

n k 1

1

[Ramanjuan] Berndt Ch. 2

=1

212 ( )s rsr

J1123

=

2 132 xx

dx

xx

dx

=dx

ex

10

=

1xe

x

e

dxe

=dx

x x

dx

x x2 3 1 320

20

1

= dxx

xdx

x

x

12

1

02 )1(

log

)1(

log GR 4.231.6

= dxx

x

122 )1(

log GR 4.234.1

= dxx

x

1

03

4 )1log(

=

4/

0

2/

0 cos)sin(cossin1

xxx

dxdx

x

x

=

03

4sindx

x

x

= GR 4.393.1 dxxxdxxx 2/

0

2/

0

tanlog)cos(tanlog)sin(

= GR 4.384.3 1

0

2cos)log(sin dxxx

=

02 )2/cosh()1( xx

dx

= dxxx

x

0 cosh

)2/tanh( GR 3.527.15

= GR 6.2111

0

)( dxxli

... 2 22 1

21

logk

kkk

=log( ) log2

21

x x

xdx

...

020 3)!(

)1(

3

2

kk

k

kJ

...

1

3

!

)(

k k

k

... )3(

...

23

2

2

)1(

21)()1()3(2

3 kk k

kkkk

... 4/34/34/14/1 )1(1)1(1)1(1)1(14

i

=

1

1

14

1)14()1(2

1

1 k

k

k

kk

k

...

1

4/1

)!2(

!

2

1

4

3

4

1

k k

kkerf

e

... /1

...

1

1

)!2(

4)1()2(2log2

k

kk

kkci

...

22 2

log

k kk

k

... 6 9 .695359714832658028149... 94

...

1

2

2

)1(2log668

kk

kk

... ie ei i

22 2

( ) sink kk

1 11

...

1

2

)!2(2sinh2

k

k

k

ee

...

0 )!2(

coshk

k

k

ee

... u k

kk

( )

2 11

... Ei

e

k ek

k

k

k

( ) ( ) ( )

!

1 1 11

1

...

1 2

1

kk k

...

2/1

2

3

16

... 21

3 33

2

21

Lix

x xdx

log

.69778275792395754630... dxxx

xLi

1

0

2

2

)1)(2(

log

2

1

3

1

18

... 11

2 122

kk

.698098558623443139815...

1

1

15

1)25()1(2

1

1

1

k

k

k

kk

7 .69834350925012958345... sinh

!( )

4

2

41

20

k

k k k

.6984559986366083598...

0 )!12(

2)1(

2

2sin

k

kk

k GR 1.411.1

=

0

1

)!2(

2)1(

k

kk

k

k

.699524263050534905034...

12 2/2

12log68

k kk

.699571673835753441499... | ( )| k

kk 2 11

=10

7

...

3/13/2 )1(44

2

1)1(1()1()1(

6

1 ii

3/13/213/23/1 )1(44

2

1)1(1)1(1)1(1

6

1

3/23/1 )1(1)1(16

1

=

1

1

133

1)36()1(2

11

k

k

k

kkk

.700170370211818344953...

122

12

11

)1(

)12(

kk kkLik

k

k

2 .700217494355001975743...

2

33 1)(k

kk

.700367730879139217733...

122

11

kk

.700946934031977688857...

1 )!2(

),2(2

k kk

kkS

1 .70143108440857635328... x dx

e xx

2

0

.7020569031595942854...

1

31

3

1

2

1)3(

x

dx

kk

.70240403097228566812... )()1()()1(4

1

2iiii

= )1()1(4

1

2

1ii

i

= 1

12 11 k k kk

.70248147310407263932...

02 552 x

dx

.702557458599743706303...

0 )2(2!

124

kk kk

e

.703156640645243187226... 2log23

8

2

5

1 .70321207674618230826...

2 12 69

4)(

3

23log

33 k k

k

kkk

...

0

2/)12( )12(

1

4

1tanhlog

2

11arctanh

kk kee

J945

...

4

1sin2log

))2/1cos(1(2

1log

2

1 GR 1.441.2

=

1

2/2/ )2/cos(1)(1(log

2

1

k

ii

k

kee

...

1

2!

22,2,2,2,1,1,1

k

k

kkHypPFQ

...

222 2

11log

2

)1(2

1)(

kk

kk kk

k

... 4G

.7041699604374744600... dx

xx1

... 2

4

...

( )

( ) ( )

( )

( )

k

k k

k

k kk k k

11

1

12 2 2

...

( )k

kk 12

...

1 )(

1

k kp, p(k) = product of the first k primes

... 11

81

k kk ( )

... ( )( )

log

11

2

11

1

21

1 1

kk

k k

k

k k k

...

0

2 )!2(!

)1(2)2(2

k

k

kkJ

...

1

202

4 )()2(

36 k k

k Titchmarsh 1.2.2

= 17

12

.7060568368957990482... 3

21 1

1

2 120

log( )x

dx

... )6(60

137

... 42

)1)(2

kk

k

.70673879819458419449...

2

1)()1(k

kk

.7071067811865475244... 2

2

1

2 4 12 12 cos cos sin

= ( )

11

4

2 1

8

2

0 0

kk

kk

k

k

k

k

k

=( )!!

( )!!

2 1

2 21

k k

k kk

=

01 36

)1(1

36

)1(1

k

k

k

k

kk J1029

= 11

2 11

( )k

k k

1 .707291400824076944916...

1

31

3

20

3 )()(

)6(

)3(

kk k

klc

k

k

Titchmarsh 1.2.9

... 1

21 2

11

2

log ( )k

kk

k

15 .707963267948966192313... 5

.70796428934331696271...

kLi

kk

k

kk

111)1(2

212

... 7

21

7

11

2 1

70

k

kk

k

k

( )

... 2/1)2(

1

1212

)!2(

2)1(

2

1cos11sin H

kk

kk

GR 1.412.1

6 .70820393249936908923... 45 3 5

...

0 13

)1(

3

2,

2

1

6

1,

2

1

6

1

k

k

k

.70849832150917037332... 2

12sin

)1(

2

1sin1cos22log

2

1(cos

2

1

1

1

k

kk

k

... dxx

x

1

0

2/11

11024

231 GR 3.226.2

.70871400293733645959...

2

11

kkk

... 2log2

92log

2

)8(324

82

2

G

=

124k

k

kk

H

... e

1 .70953870969937540242... 2 3

20 1

120

e k

kk

, ,( )!

.70978106809019450153... 8 2

1 1 21

1 2 1 22

40

( ) ( )cos( ) ( )

i e i ex

xdxi i

... 3/15

...

12)1(

2)2(24

k kk

= 1

0

2 )1log()log( dxxx

.7102153895707254988... sinh2

1201

164

3

kk

... dxx

x

0

2

cos12log4 GR 3.791.6

.71121190491339757872...

12/)3(2/)3(

122

2

1

2

1)1(

kkkkk

k Hall Thm. 4.1.3

3 .7114719664288619546...

31 1)(

kk kF

... )3()2(22

)5(9)2,1,2(

MHS

... coth

42

3

... 2

13tan

133

4

... 1)()()(2

0

kkkk

...

22

)(

1 1 22

)(2

1

221

)4(

)2(

k

k

k s ks

k

kkk

k

...

1 3

)1(14log3log1

kk

k

k

...

22

1log

1

11)()1(

kkk

k

k

k

k

k

kkH

... 3

24

1 1

12

12

1

( )!( )!

( )!( )!

k k

k kk

...

0

3

3

0

2sin2

)1(2

2

3

x

x

k

kk

k

k

... 4log7

...

03 12

)1(

k

k

k

...

0

22

)!2(1

211

2coth

2 k

kk

k

B

e

J948

=

0 1

2/sindx

e

xx

...

12 14

21

2coth

2 k k

J948

...

3

2

3

1

2

3log33log3

2

ll Berndt 8.17.8

... cos ( ) sin ( ) sin1 1 1 1 1si ci

= log cos( )x x dx10

1

.71317412781265985501...

( / )

( / ) ( / )

( )1 2

5 6 1 31

1

31

k

k k J1028

... 2

2

0

1

16

2

2

1

4

logarctanx xdx

1 .71359871118296149878...

1

)13(k

k

= 7

5

... G2

...

12

2

2kk

k

k

H

...

1

02 )21(

arcsin

3

31log

2dx

xx

x

... 1

0

1

02

sinh1

cosh

4

1

4

3dx

e

xdx

e

x

e xx

...

222

11

2sin

22

k k

...

0

3/1

3!

)1(

kk

k

ke

.716586630228190877618... dxx )41log(22arctan5log1

0

2

... 27

... dxee

eexx

x

1

0

2

2

1log

2

1

...

22

1)(

kkk

k

8 .7177978870813471045... 76 2 19

... 33

32 CFG G1

.7182818284590452354... ek k k k kkk

21

3

1

1 200 !( ) !( )( )

=1

1 20 ( )! !k kk

1 .7182818284590452354...

0 1

2

)2(!)!1(1

k k kk

k

k

ke

2 .7182818284590452354...

0

/2

!

1

k

i

kie

= dxex

x

02)/1(

log

3 .7182818284590452354...

0

3

)!1(1

k k

ke

...

2

12

1log2

csch2

logii

=

1 12

1

2

11log

2

)2()1(

k kk

k

kk

k

... 22

1

3

1

2

1

4

3log

2

3log2log

12 22

22

LiLi

=

0 2xe

x

... dxe

xx

02

3

)1()4()3(6

...

13

1 133

)13(

kkk k

kk

.71967099502103768149...

2 !

)(

k k

k

... 11

48 12 50

log

( )

x

xdx

...

12 244

1

2tanh

4 k kk

J950, GR 1.422.2

= dxx

x

0 2sinh

2sin GR 3.921.1

=

0

sinxx ee

x

... H

kk

kk

( )3

1 2

... )2(2)3(5

...

0 )3/1)(1(

)1(

2

3

k

k

kk

.72072915625988586463...

02 5

1

2

19tanh

19 k kk

... 4

arcsinh

... 5

)3(3

... e4log4log

1 J153

8 .72146291064405601592... 22

k

k

k( )

...

1

1

23

)1(3

2arccsch

7

3

7

4

7

3

k

kk

k

k

... log

=13

181

1

1

2

40

log( )( )

xx

xdx

... )!1(

1

2

3

2

2

1

2/3

k

e

k

k

... 21 3

16

2

8

1

1

2

20

1 ( ) log log( ) log( )

x x

xdx

1 .72257092668332334308... 18

3

...

222 )13(

3

3

1)(

2

333log9

kkk kk

k

... dxx

x

0 sin2

sin

33

4

.72330131634698230862...

22 )(

1)(

k k

k

...

1

3

)!1(

3

3

12

k

k

k

ke

...

13

12 1

)1(4

)3(3

12 k

k

k

k

.72475312199827158952... ( ) log sin3 2

8 30

x x

xdx

1 .724756270009501831744...

1

K

4 .7247659703314011696... 105

16 3 , volume of the unit sphere in R7

.724778459007076331818...

1

1

1

1

2!

2)1(

!)!12(

)1(

2

1

2 k

kk

k

k

k

kk

kerfi

e

.72482218262595674557... 1

2

2

3 3

2

3

1

3 112

1

cot( )k

kkk k

1 .72500956916792667392... dxxx

x

0

22

)1)(2(

log

6

2log GR 4.237.3

.72520648300641149763... 150

97

40

, probability that three points in the unit square form an

obtuse triangle

.725613396022265329004...

11

)1log(

kkk

e

H

e

eee

... 2/

0

)tan3log(2

3log dxx GR 4.217.3

...

2

1

2

2

0

tanh sinsinh

x dx

x

.726032415032057488141... 3

1

8

... log log2 23 2

...

0 2

2

2cosh

2sinh

)1log(4

2 dxx

x

x

GR 4.373.5

18 .726879886895150767705...

2

3 )1()()3(6)2(7k

kkk

=

232

23

)1(

1458

k kk

kkk

.72714605086327924743...

1

22

22

2

2sin

2 k

ii

k

keLieLi

i

... )4()2(

= 11

8

... 1

2

2

42

0

2

sin

cos (cos ) sin/

x xdx

...

0

/1

!

)1(

kk

k

ke

...

5

32Li

.72760303948609931052...

2

312

7

23

1 2

( )( / )

k

kk

1 .72763245694004473929...

1 2!

11

kkk

.72784877051247490076...

2 !!

11

k kk

... sinh

1

... 2

.728102913225581885497... dxxx

x

124

2

14

3log

34

...

... log ( )212

21

2

3

23

2

3

Li

= log

( ) ( )

2

30

1

1 2

x

x xdx

1 .728647238998183618135... 2)( xofroot

... 7

30 1

4 4

21

log

( )

x

xdx

.7289104820416069624... 21

64 2 14 30

dx

x( )

... cosh

1

.72908982783518197945... 45 5

64

4

31

( ) log

x

x xdx

... 221 e

... e )2sinh2sinh(cosh)2sinh2cosh(cosh2

1

=

0 !

sinh

2

1 2

k

ke

k

keee

1 .7299512698867919550...

2 1

2

cosh(log ) cos( log )i

.73018105837655977384... 22

12

0 )12(2

1

2

22

28

2log

kk

kGk

k

k

Berndt 9.32.4

= 4/

0

cot

dxxx

= 4/

0 2cos

tan4/ dx

x

xx GR 3.797.3

= 4/

0

)sinlog(cos

dxxx GR 4.225.1

= 4/

0

)cottanlog(

dxxx GR 4.228.7

=

1

02 )1(

arctandx

xx

x GR 4.531.7

1 .730234433703700193420... dxe

eerf

ex

x

0

4/1

2

2

11

2

.730499243103159179079...

3/2

3/1

.731058578630004879251...

0

)1(2

2/1tanh1

1 k

kk ee

e J944

... 2

27 130

log x

xdx

= dxee

exxx

x

03 )1(

)1( GR 3.411.26

...

2

)1)()1(k

k kk

1 .73137330972753180577...

2 21

1

kk

1 .73164699470459858182... 3

...

1 2!2log

2

1

kk

k

k

HEie

.731796870029137225611...

2233

1 )1(

3)1)3(3

kk kkkk

2 .731901246276940125002...

1

7

/1

7 )!7(

)(

k

k

k k

e

k

k

.732050807568877293527... 3 11

2 1

2

0

( )

( )

k

kk k

k

k

1 .732050807568877293527...

0 0

2

6

2

6

1

3tan3

k kkk k

kk

k

k

... 12

5tan32

AS 4.3.36

21 .732261539348840427114...

04

442

)1(

log

45

7

3

2dx

x

x

... 1

... 1

...

2

1

2

12log

2

1

8

2log3 22

iLii

iiLiG

=

12

2

11

1log

x

dx

x

x GR 4.298.14

...

22 1

)(

k k

k

.733333333333333333333 = 11

15

=

1 1 )3(4

12

)3()!2(

!)!12(

k kk kk

k

kk

k

...

2232

1 )1(

11)13(

kk kkkk

...

1

,1

11csc

...

1 0 )(2

1

kk k

... 16 16 22

32 3

2 12

3 121

log ( )( )!

( )!

k

k kk

... 2 12

12

1

( )!

!

k

k kk

... e

x xdxecoslog coslog

1

2 1

.73462758513149342362... 2

2 21

1

42 41

sin sinh

kk

... 3

4

CFG G1

...

1

3

)5(!11033000

k kk

ke

... ( )k

kk 21

...

1

22

12

/1

)1(1csch1

2 k

k

k

.73575888234288464319...

1 1

5123

!

)1(

)!1()1()1,2(

2

k k

kk

k

k

k

kk

e

3 .7360043360892608938... 2 11

21 4

40

/ log

xdx

...

1

2 )1)1(k

k kH

.73639985871871507791...

11 21

4!)!12(

!)!2(

3

1

27

32

kkk

k

kk

k CFG F17

=cos

( cos )

/ x

xdx

2 20

2

... ( )( ) ( )

11 1

2

k

k

k k

k

...

1 22

/1 2

)!1(

1)2(

k k

k

k

e

k

k

= 19

14

... )3(3 ... ( ) ( )( ) ( )2 2i i

... ( ) /k

kk

kk k

1

2 3

1

2

3 2

2

arctanh

...

0

22

4/sinh

4/cosh)1log(12log

281624

x

xx

GR 4.373.6

1 .7380168094946939249...

1 1

22

3

)(

13k kkk

kk

...

3

23Li

... p

kk

k !

1

...

2 6

)(5)1(

3

2log5

4

3log5

34

51

kk

kk k

.73879844144149771531...

1

1

2

)1()1(

2

3log

3

1

kk

k

k

k

.7388167388167388167 =

2

1,6

693

512

.739085133215160641655... xxofroot arccos

.7391337000907798849...

13

1

33

2 sin)1()()(

212

1

k

kii

k

keLieLi

i Davis 3.34

19 .739208802178717237669... 22

... 1

1

1

1

1

12k kkk

j

j kjk

( )

.740267076581850782580...

12 3

1

6

13coth

6

3

k k

J124

7 .740444313946792661639...

1

4

21010 !!)2()2(3)2()2(2)2(2

k kk

kIIIII

... 23

CFG D10

.740579177528952263276...

3

21

2

2

)()1(

24

52log

4

)3(7

kk

k kk

=

132

2

)12(2

21118

k kk

kk

=

1

2

427

20

kk

k

1 .740802061377891359869... 31log3

...

3

1

3

)1(

3

1

6

3

2

3log3

112

hgk

kk kk

k

= dxx

x

1

02

3)1log(

... dxe

xx

1

0 1

sin

.741227065224583887196...

122

2

)2/1(

1

2coth2

32

k kk

...

22

3

)1(24

2log5

8

7

kk k

k

... 14

...

2

1)( 1)1(k

kk e

...

2

2 2)1()(k

kkk

... 22

2 10

1

Gx

x xdx

log arcsin

( )

=arccosx

xdx

10

1

GR 4.521.2

=x x

xdx

sin

cos

/

10

2

GR 3.791.12

=log( )1

1 20

1

x

xdx GR 4.292.1

... pf k

ek

( )

0

...

1

21

)()2(log

)()2(

kk k

kk

k

k Titchmarsh 1.6.2

.744033888759488360480...

1 )12(2kkk

k

2 .744033888759488360480...

1

1

11 2

)(

222

1

12 kk

kkk

kk

kk

=

1 )12(2kkk

k

... 2

)3(G

...

1

3

3 5/sin

125

3

k k

k GR 1.443.5

.74430307976049287481... cos/

x dx0

4

.74432715277120323111...

1

1

12)1(

2

1arcsinh

55

8

5

2

k

k

k

k

...

log

5 .74456264653802865985... 33

... = 2

204 2 2 1

log xdx

x

... e

e

8

1

8

5)1sinh21cosh3(

4

1

=

1

2

)!12(k k

k

... )(log)(2

kkk

... 5

3

1

5

2

0

( )k

kk

k

k

... 1sinsin

... 1)2()(1

kkk

7 .7459666924148337704... 60 2 15

... e8

...

1

6

/1

6 )!6(

)(

k

k

k k

e

k

k

9 .7467943448089639068... 95

...

21

1

2 10

erfk k

k

k

( )

!( ) J973

= e dxx2

0

1

... 2

32 2 4

1

2

22

2

2

12

20

1

loglog

( )Li

x

xdx

.747119662029117550247...

( ) ( )

( )

k k

kk

1

2

2 .747238274932304333057... 1

22 5 15 6 5

= 5 5 5 5 5 5 5 ... [Ramanujan] Berndt Ch. 22

... 59535 7 62

24 1

6 7

30

1 ( ) log

( )

x x

x

... G3

...

1

2

!

2)2(2log

k

kk

k

HEie

... )2()2(2

1coth

22 ii

= )1()1(2

1iii

i

= 2)12()2()1(1

1

1

23

kkkk

k

k

k

k

...

2 1

128

1 3 1 1

4G

kk k

kk

( ) ( )( )

... 8

7

...

1

41

4

1

3

1)4(

x

dx

kk

.74912714295902753882... 3 3

3

1 21 12

1

( ) ( )!

!

k k

k

k

k

... 2/e

...

1 )1(2

1

kk k

.750000000000000000000 =

1 3

)2log(sin4

3

kk

kii

=

22

1 1

11)2(

kk kk J369, J605

=

12 2

1

k kk K133

=

13

2)22()2(

2

1

kk

kkkkk

= 11

3 21

( )kk

=

13

4log

x

dxx

1 .750000000000000000000 = 7

4 21

H

k kk

k ( )

=11

4

1

1

2 4 6

2 6 8

p p p p

p p pp prime

8

372 .750000000000000000000 =

1

6

3kk

k

.75000426807018511438... 4

33

1

21

20

1

arctan tancos

cos

d

...

1 )()()1(

k ekk

eehge

... 2

22

1

20sintan

kk

k

...

2

3

4)( log

)3(k

iv

k

k

... 4/1

... 5 3

8 2

2

1

( ) ( )

H

kk

kk

=

log( ) log( )1 1

0

1 x x

xdx

... dtt

kk

k

0

2

13

13

4tanlog

)2/1(

)1(

4


...

1

22

1

22 1

2

121

log ( ) ( )! ( )e

Eik k k

kk

k

... HypPFQk k

k

k

{}, , , ,( )

( )!( )!

1

2

1

21

1

16

1

2 20

...

14

2/1

0 2!

)4(

k

k

kk k

e

k

k

k

k

1

2

2

2

3

1

27

34

36

1 )2(3

=

2

31

2

31

3

1

2

)3(33

iLi

iLi

2

31

2

31

3

333

iLi

iLi

i

= 1

06

2

1

log)1(

x

xx GR 4.261.8

... 23

96 12 40

log

( )

x

xdx

... )4(34

... )!1(

2

)!1(

122)2(

001

2

k

k

k

k

kk

kIe

kk

122

11

k kk

... e Ik

k kk0

1

1

21

2 1

2

( )

( )! !

!!

... e I e dxx0

0

11

2

2

cos

...

22

1

21 1 2 1 2coth i i

= 2 1

21 2 2 2 13

1

1

1

( )( ) ( ) ( )

k

k kk k

k

k

k

k

... 15

25 2 3

1

3

21

( ) ( ) ( )( )

H

kk

k

(conj.) 18 MI 4, p. 15

.754637885420670280961...

e

xe

dxxee

e

0

11

... 6

2

31

126

0

1

loglogx

xdx

... 3log3 2

...

2 21

1 120

log( )

( )(

k

)k k k

...

2

15

2

15log

10 22

2

Li

Berndt Ch. 9

1 .75571200920378862263...

1

/1

1

1!

)5( 5

k

k

k

ek

k

... sinh2

2 412

14

kk

...

0 )!2(k

k

k

B

...

0 )1(2)2log(2

x

dxEi

x

.75735931288071485359... 235

.7574464462934689241... 2 5 2 3 34

2 1

2

41

( ) ( ) ( ) ( )

( )

( )

k H

kk

k

... 1

120 e

k

k

... 9

4)3(

.757672098556372684654...

1

22

)12()2(8

1

2

)3(

8 k

kkk

=

223

2

)1()1(

1

k kk

k

... )()22()()22(4

1

4

)3(333 iLiiiLii

= dxxx

x 21

04

log1

1

...

0

)1(

1 kk

k

... 11 2 2 1

0

1

eEi Ei e e x dxx( ) ( ) log log( )

...

3

2

2

3log3

6

3

2

3hg

... 2 3 2 ( ) ( )

...

1

2 1)2(1)2()3(2k

kk

... x dx

e ex x

1

...

... 4/13

... 2/2

2/2

2

2

1

16

1

12ii eLieLi

= 2

cos)1(

12

1 k

kk

k

.75997605244503147746...

1 12 )!(

1

!

)(2

k jj

k kk

ke

.7602445970756301513... cos( )

( )!

1

2

1

2 20

k

kk k

AS 4.3.66GR 1.411.3

.7603327958712324201...

1 5

11

kk

...

1 )!2(

),2(1

k k

kkS

.760345996300946347531...

1 16

)1(

16

)1(1)32log(

3

1

k

kk

kk J83

=

1

1

1 22

)1(!)!12(2

!)!2()1(

k

kk

kk

k

kk

kk

k J267

=

0 !)!12(

!)1(

k

k

k

k J86

=

22 3x

dx

1 .76040595997292398625... 16 232 2 1

16 1

2

4 30

log( )!

! ( )

G k

k kkk

...

1

2

5

10 )!()2(4)2(5

k k

kII

...

2

1 1)1()()1(k

kk

.76122287570839003256... cot2

...

13

1log

1

k k

k

k

...

4

335

4

33

4

35

4

33

32

11

iiiii

=

02 1

)1(

k

k

kk

... u k

kk

( )

!

1

...

2

/1

1

1

!

1)1(

k

k

k k

e

k

k

.761390022682049161058...

2 !

)(

k k

ke

...

0

2sinh)1(4

2

dxxee x

... )1(log

... iie

etan

1

11tanh

2

2

J148

=

1

2)1(21k

kk e

=

1

2

)!2(

)14(4

k

kkk

k

B AS 4.5.64

= 1

02cosh x

dx

.761747999761543071113...

03

3/1

233

2

x

dx

... 2

3sin

...

1

1

!!

)1(

k

k

k

...

( ) ( ) ( )( / )

1 33 2

41

1

3

1

54

1

3

4

9 3

26

33

1 3

k

kk

2 .76207190622892413594... 2 3

30

12

3

2

6

1

2

1 2log log log( ) log

Li

x x

xdx

... cosh( )

22

4

2

2 2

0

!

e e

k

k

k

AS 4.5.3, GR 1.411.2

=

0

22 )12(

161

k k J1079

2 .762557614159885046570...

1

71

2!)1(

128

583

kk

k

k

k

e

...

2

0

2

)()(

kk

kk

1 .762747174039086050465... dxx

x

e

dxx

0 1

2

arcsinh

11arcsinh221log2

...

44

)3(

4

)()3)(2)(1(

4

3

kk

kkkk

...

1

3

2)!1(4

32

kkk

ke

... 2 1 1 11

2 10

sin cos( )

( )!( )

k

k k k

...

4

3

4

12log72log6 2 ll Berndt 8.17.8

... 53

.76410186989382872062...

12

2

)4/3(

1

9

168

k kG

...

2

)(1)(k

kk

.764403182318295356985...

1

42

42

4

4 sin

4

1

180 k

sii

k

keLieLi

.764499780348444209191... 1

2 1

1

2 11

4 1

2 4 11

1

1

12k

k

k

kk

kk

k k

( )

( )( )

( ) Berndt 4.6

=

1 )12(2

11

kkk

...

1

2

20

24 )(

)4(

)2(

72

5

k k

k

Titchmarsh 1.2.10

1 .764667736296682530356... )4()3()2(

= 17

13

.7649865348150167057...

sin sinh/ /

3 3

2 31

31 4 1 4

2 42 k

...

0 02200 )!)!2((

)1(

4)!(

)1()()1(

k k

k

k

k

kkiIJ

...

1

3

3 3/2sin

81

2

k k

k GR 1.443.5

... 11

1 21

k k kk ( )( )

... 1)()1( 5 kkk

... 3

2

5

3 20

log/

dx

ex 3

...

1

02)1(

)2()1(1

21 dx

x

eEiEi

e

e x

... 11 13 2

3 31

coth( ) ( )

csch ik i k ik

=

0

2

1

sindx

e

xxx

...

1 )!1(1)1(12

k

k

k

kHEi

...

1

2)2(logk

k

1 .767766952966368811002... k

kkk

k

k

k

k

kk

8

2

2!)!2(

!)!12(

22

5 2

01

2

1 .76804762350015971349... 3/23

3/13

3

)1()1(3

2

381

8 LiLi

i

=

03

21

02

2

1

log

1

log

x

dxx

xx

dxx GR 4.261.3

5 .7681401910013092414...

12 1

21

k k

.76816262143574939203... 61 3

86 6

21 3

23

3 3Gi

Li i Li i

( ) ( )

( )

=arcsin3

20

1 x

xdx

... 13

82

4

2

22

( ) coth( ) k

kk

= ( )2 11

16 2

2

kk kk k

... 2 22 12 1

1

/ ( ) /

k k

k

...

0

33/2

cos3

1

26

13 3

dxxe x

... =

112

11

kk

... 3G

.7687004249195908632... =1

20 ( )!! !!k kk

... cos ( ) cosh ( )( )!

/ /2 1 2 12

41 4 1 4

6

0

k

k k

2 .768916786048680717672... arcsinh12

1

2

20

log( )x

xdx

...

2

0 cos24

cos

3

21 dx

x

x

... i2Re2logcos

... 2 2 4 6 2

0

1

log arccos logG x x dx

31 .769476621622465729670...

1

4

32

3log15

61

411

kk

kHk

.769837072045672480216... 22

222 csch4

2sinhcsch8

cschcoth4

1

=

1

1

2

1)2()1(4

1)2()()1(

k

k

k

k kkkkk

=

122

2

)1(k k

k

... 8

7)2(

.76998662174450356193... 2 2 11

1 211

0

Jk k

k

kk

( )!( )!

.770151152934069858700...

0

2

)!2(

)1(

2

1

2

1

2

1cos

k

k

k J777

.770747041268399142066...

0 2!)1(2

1

kk

k

k

B

e

... 4

11

4

2

22

2

14 1 2

2

20

eK

x dx

x/ ( )sin

= GR 3.716.9 2/

0

2 )tan2(sin

dxx

.771079989624934599824...

1

2

4

)2(

kk

k

... 3

231

k

k k

kk

1 .771508654754528604291... )3(4)2(4

.771540317407621889239... 2

1cosh

1 .7720172747445211300...

12 1k k

k

.772029054982133162950...

sinh2

1

.77213800480906783028...

2 2

1

2

1

2 11 2

40

ex

xdx

/ cos sincos

1 .772147608481020664137... 2log12

)3(2

...

1

2

=( )

( )

k

kk

!

!

1

2

1 1

... 4 2 22 1

2 1 21

log( )!!

( )! ( )

k k

k kk

=

22

0 2

1)(

)2(2

1

kk

kk

k

k

=1

1120 ( )(k kk )

= dx

e ex x

1 20 /

= 1

0

1

0 1dydx

xy

yx

...

1

1

212log4

kk

kH

... )1(4

112

2

)1(2log4

01

kk

kHkk

kkk

k

=

0 )2/1)(1(

1

k kk

=

222

)(

kk

k

=

03

4 2sindx

x

x

... dxx

x

1

0

2/9

1256

63 GR 3.226.2

... 1 1

12 p p pkp primep primek

( )

= )(log)(

2 1

skk

k

s k

1 .773241214308580771497...

1

2

)!2(

22sinh22cos2

4

1

k

k

k

k

... 7

3)3(

... 4log6

...

1 1

1

k kk FF

... )sin( e

...

0

4

!15

k k

ke GR 0.249

... 3

264

1

1 4

1

1 4

3 1

3

1

31

( )

( / )

( )

( / )

k k

k k k

... 1

2 0

sinh sin sinh

x x dx

... 3

5

1

6

2

0

( )k

kk

k

k

1 .774603255583817852727...

1 82

2224464

kk

kk

...

0 2

)1(

kk

k

k

...

2

1

12

5

6

3 1 1

62

( ) ( )( )( )

k

kk

kk

8 .7749643873921220604... 77

...

1

1

0 !!

)1()2(1

k

k

kkJ

...

21

1

kkk

k

3 .776373136163078927203... )3(

... dx

x

x

20

1

2 .77749955322549135074...

215

/1

)!5(

)(

kk

k

k

k

k

e

... 2

1)1(

1)1log(1

6 22

2

eLi

eLie

=

02 )!1(

11

k

k

k

B

eLi

=

1

0 1xe

dxx

= 9

7

17 .77777777777777777777 = mil/degree

14 .778112197861300454461...

0

12

!

22

k

k

ke

... 4/110

1 .77844719779253552618... pf k

e kk

( )

!

0

... tan ( ) ( )

( )!

1

2

1 2 2 1

2

1 2 1 22

1

k k k

k

k

B

k

.778758579552758415042...

11

1 1cos1

)!2(

)2()1(

kk

k

kk

k

.778800783071404868245...

0

4/1

4!

)1(

kk

k

ke

1 .778829983276235585707...

02

2

2 1

coslog

1

2log

x

x

e GR 4.322.3

1 .779272568061127146006... )3(

...

3 3

2

3

2

3 10

1

arctan

e

e

dx

e ex x

...

1 )3(

11

k kk

... 3

tanh

.7808004195655149546... k

ekk

11

...

1 2

)(

kk k

k

...

0 !k

k

ke

=nn

n

loglog

)(lim 1 HW Thm. 323

.781212821300288716547... )1(1Y

.781302412896486296867...

122

)1()1(2 )13(

1

)23(

1

3

2

3

1

9

1

k kkg

J310

= dxxx

x

1

021

log GR 4.233.1

.78149414807558060039... 2

221

k

k

k

.781765584213411527598... 2

3log3

3210

21

3

8

6 .781793170552500031289... )2()(13

)3( 22

kkk

=

223

23

)1(

124

k kk

kkk

1 .78179743628067860948... 2 11

6 55 6

1

1

/ ( )

k

k k

.781838631839335317274...

1

2/1 12k

k

1538 .782144009188396022791...

4

1)3(

113 .782299427444799775042...

1

2/2

!

sinh2

2

1

k

kee

k

kee

6 .78232998312526813906... 46

1 .7824255962226047521... 1

0

2log]1},2,2,2{},1,1,1[{2

xe

dxxHypPFQ

.78245210647762896657426... F kkk

( )2 12

... 1

6

8 2

9

33

1

log log( )x

xdx

...

1

1)1(

kk

k

k

...

2

1)(

k kF

k

=

1

41

2)1(

125

598

kk

kk kF

2 .784163998415853922642... 2

2/3

.7853981633974483096...

0 12

)1(

2

1arcsin)1(

4 k

k

k

AS 23.2.21, K135

= )(Im1log(Imarctanh 1 iLiiii

=

1 )14)(14(

121

k kk GR 0.232.1

= ( )

( )sin( )

1

2 12 1

1

21

k

k kk J523

=

1 12

)12sin(

k k

k GR 1.442.1

=

1

2/sin

k k

k

= sin3

1

k

kk

GR 3.828.3

=( ) cos( ( ))

1 2

2 10

k

k

k

k

1 / 2 Berndt Ch. 4

= arctan( )

2

2 2 20 kk


= arctan1

2 21 kk


=

12 1

1arctan

k kk K Ex. 102, LY 6.8

= ( ) arctan

121

21

k

k k [Ramanujan] Berndt Ch. 2

=

1

2)12(

11

k k GR 0.261, J1059

=k k

kk

( )

( / )

1

1 2 21

J1061

=

02

02

02

022 14224)1( x

dx

xx

dx

x

dx

x

dx

=

041 x

dxx GR 2.145.4

=

03/24

023 21 x

dx

xxx

dx

= x dx

x1 40

1

=

log

( )

log

( )

x

xdx

x

xdx2 2

02 3

01 1

=

0xx ee

dx

=sin

sin

/ x

xdx

2 20

2

=sin4

20

x

xdx

=

0

sin1 dx

x

xe x

= dxxx2/

0

2 tanlog)(sin

= 1

0

logsinlog

)1(dxx

x

xx GR 4.429

= GR 4.533.1

0

)arccot1( dxxx

= GR 6/469/1 1

0

cossin)( dxxxx

1 .7853981633974483096... 4

12

2 30

k

k k

k

.78569495838710218128... cos sin( )

16 161

1

8 41

k

k k J1029

.7857142857142857142 =14

11

.78620041769482291712... 8

9

8

27

1

2 2 arcsinh

= 2 10

111

2

1

8

12

2

Fk

k

k

kk

, , ,( )

...

212 2

log

)!1(

1

2

log

kk

n

nkk

k

n

kk

...

1

4 1)2(k

k

... 1

11

112 1

1 ( ) ( )k k p pkk k

kp prime

... )2sinh2sinh(cosh)2sinh2cosh(cosh2

1e

=

0

/1

!

sinh

2

1 2

kk

e

ek

kee

.786927755143369926331...

1

51

5

1

4

1)5(

x

dx

kk

.78693868057473315279... 22 1

1 20

e k

k

kk

( )

( )!

... ( ) ( )log

( ) ( )2 2

1

2

3

23

1 23

2

20

1

Lix

x xdx

...

12/ 1

1

kke

Berndt 6.14.4

...

0

3

)!2(

3

9

4

k

k

k

e

... !5log

... csch

24

2log

2coth

42

1 2

2

12

12

1

2

1

4

1 iiii

=

)1(

1)1(

21

kk

kk

... 2/1)3()3(68 H

...

2 )(

1

k klfac , )...,,1()( kLCMklfac

...

1

3 1)2(k

k

... 15 5

25 4

2

( )( ) ( )

1 .78820673932582791427...

3 122

0 )(log)(log)(log)(n kkk

knkkk

... log log sin( )

( )!( )2 2

1 4

2 2

12

2

1

k

kk

k

B

k k [Ramanujan] Berndt Ch. 5

= 1

0

2

x x

xdx

cos [Ramanujan] Berndt Ch. 5

.788337023734290587067... coth

4

.7885284515797971427...

( )( ) ( )

( )

3

2

3

161 2 1 1

2

12

3

2 32

k k kk

kk k

...

22

22

log11log

1

1

1)(

kkk kk

k

kkk

k

=

1 21 2

111)(

n kn

n kn kk

Lik

k

... k

kk

k

1

12

( )

... 5

4

... 3log8

= 19

15

... =

1

1 !!

2222

k kk

kI

... ( )( )

log

11

2

k

k

k

k

...

1

2

43

4log44

81

188

kk

kHk

...

1 2

1

k kk H

.7907069804229852813... 1

1 2

1

21 0 1( ) !!

!!

( )! ( )!!k k

k

k kk k k

!!k

... H

kk

k

3

31 1( )

1 .791759469228055000813...

11 6

5

6

56log

kk

k

kLi

...

1

2

1

2/1

)1(1

2csc

2 k

k

k

1 .792363426894272110844... )1()1()(4

coth22

)1()1(22 iiiii

=

12 1k

k

k

H

.792481250360578090727...

1

4 222log2

3log3log

xx

dx

... 2

7

...

1 2

1

kk ke

...

2

2

.7939888543152131425...

( / )

( / ) ( / )

( )1 2

9 8 3 81

1

41

k

k k J1028

.794023616383282894684... e2 Berndt 8.17.17

.79423354275931886558...

0

)1()1()1(

1

sin)1()1(

2)(Im dx

e

xxii

ii

x

... kk

Hk 21

1)1(

... 1

42

20 0 2

1

I e Ie

k k

kk

( )sinh cosh

( !)

... )3(

3

...

0

3

)2)(1(2202log30

kk kk

k

.795371500563939690401... 5/1

.7953764815472385856... HypPFQ

kk

k

k

k

[{ , , },{ , }, ]( )

( )

1111

22

1

4

1

120

... 2

sin

.795749711559096019168... Lie

Lie2 3

1 1

.7958135686991373424... tan tanh1 1

4 .7958315233127195416... 23

...

1

2

3kkkH

.79659959929705313428...

Eik k

e

xdx

k

k

x

( )( )

!1

1 11

1 0

1

J289

= log x dx

ex0

1

= ( )

!( )

( )

( )! !

1

1

1

120

1

1

k

k

k

kk k k k

= HypPFQ[{ , },{ , }, ]1 1 2 2 1

8 .7965995992970531343... 8 1 12 11

4

1

Eik

k kk

k

( ) ( )( )

! Berndt 2.9.8

.796718392705881508606...

1

3 1)3(k

k

...

2

1)2(

)(

k k

k

... dxxx

x

0

222

)2)(1(

log2log

9

2log GR 4.261.4

.797267445945917809011...

112

)1(1

3

2log

2

11

kk

k

k

...

2

2

)1(

log

k kk

k

... 2

2

...

22

3)2()()1(

1)3(2

k

k

k

kkk

k

9 .7979589711327123928... 96

... 3 2 3

1

120

tanh

k kk

= 13

3

3 3

2

3 3

2

i i i

...

1 )!1(k

k

k

F

...

3

1 )(!

)1()1(

k

k kk

kk

.798632012366331278403... e

k k k

k

k

2

0

1

8

2

1 3

!( )( )

... )1(

1()1(2)2(1 2

=

2

2

2

1)()(

1

k

k

k

kkk

...

1

1

)1()!2(

4)1(2sin

2

2cos

2

3

k

kk

kk

.7989752939540047561... 2 1 1

1

2

120

sin ( )

( )!( )

k

k k k

... ( )

log2 1

2

11

1

21 12

k

k kkk k

k

...

2 12 22

)(

2

3

k jk

j

kk j

kkk

... 2

2log

sinh44

1

2

1

2

1

21

21

8

1 iiii

=

023 1

)1(

k

k

kkk

...

41 24

0

2

e xcos( tan )/

dx GR 3.716.10

.800000000000000000000 = 4

5

1

41

( )k

kk

J352

= F k

kk

2

1 4

.800182014766769431681...

1112

1

2

1sin

1

2)!12(

)2()1(

kkk

k

kkk

k

1 .8007550560052829915... log( )k

kk

12

1

.80137126877306285693... 2 3

3

( )

.801799890264644999725... 7 3

8

1

4

2 1

4

2

1

( ) ( )

k kk

k

12 .80182748008146961121... log9!

... ( )32

5

.80257251734960565445... 1

5

14

5 52

12

1

1

1

arccsch( )k

k k

k

... ( )

!

41

1

1

4k

kek

k

... 3 3

2 31

1

3 22

sin

kk

.8027064884013474243... si

k k

k k

k

( ) ( )

( )!(

2

2

1 4

2 1 2 10

)

1 .8030853547393914281... 3 3

22 3 2

1

1 30

( )( )

( )

( )

k

k k

=log2

0

1

1

x

xdx

GR 4.261.11

=x x

xdx

log

( )

2

20

1

1 GR 4.261.19

=log2

21

2

0 1

x

x xdx

x

edxx

= dydxxy

xy

1

0

1

0 1

)log(

.8033475762076458819... 1

2 211

kk

... iLi e Li ei i

2 22

22 / /

... ( ) /

1 1

1

1k

k

ke

... 6 3 3

... 3

212

3

... logRe log

5

22 i

... 11

21

kk k

... e

k

k

k

1

41

4/

... ( )

!

/3 1 1

1

1

1

3

k

k

e

kk

k

k

... 1

4 2

1

4 2

1

4

22

2 2

csch csc ( ( )h i

2 4 2

2 2

( )( ) ( ) ( )

ii i

4

i

=H

k kk

k ( )21 1

... 11

61

kk

... Lkk

k1 2

1

1

2

1

2/

.8061330507707634892... 4

9 3

121

kk

kk

= 4

3

5

3

2

1 3 2 31

k k

k kk

( )

( / )( / ) J1061

=

023 )1(x

dx

1 .8061330507707634892... 14

9 3

121

kk

kk

=dx

x( )3 20 1

... 2 2 2 2 3 3 2 2 3 3 2

1

32 2

2log log log log log log Li

= log log( )x x d 20

1

x

... dx

e x

x dx

e xx x

0 0

1 .80649706588366077562... k

kk !

10

... 2 2

2 21

1

2 2 21

sin

( )

kk

... 128 32 4 8 48 22 G log

=log( ) log

/

13 4

0

1

x x

xdx

... 21

21

/k

k k

...

22

2 )2(

k k

k

... 11

22

( )k

kk

2 .80735492205760410744... log2 7

... 4

3

1

3

1

31

3

0

1

, e dxx

... 1 3

2 .80777024202851936522... dx

x( )0

Fransén-Robinson constant

.80816039364547495593... ( )

( )

k

kk

1

2

2 .8082276126383771416... 4 3 2 ( )

= ( ) log1

1

2

0

1

x x

xdx

4 .8082276126383771416... 4 3 ( )

= log( ) log/1 1 2

0

1

x x

xdx

= dydxxy

yx

1

0

1

0

22

1

)log(

5 .8082276126383771416...

1

2 2)2()1(1)3(4k

kkk

.80901699437494742410... 2

1 5

4

.80907869621835775823... 2 21

21

log( )

k

kk

.8093149189514646786... 1

2

1

2,

... 1

3

3

2

1

3 1 11 3 2 3

cosh( ) ( )/ /

=1

5 3

2

5 3

2

i i

= 11 3

32 1

k

k

kk k

exp( ) 1

... H kk

kk

( )

1

21

.81019298730410784673... 1

0

42 log)1log())6()5()4((244120 dxxx

7 .8102496759066543941... 61

.81040139440963627981... 1

22

2

1

2 120

csch ( )k

k k

... 4

1 k

k

k

... e e i i / log2

.81056946913870217... 8 4

3 2

2 2

2 12 21

21

1

( )

( ) ( )

( )

k

k

k

kk k

... 36 2 31

1 6

1

1 62

1

2

1

21

( )

( / )

( )

( / )

k k

k k k

.8108622914243010423... sin

1

11

22 kk

.810930216216328764... 2 3 21

211

1

2 10

(log log ) , ,( )

( )

k

kk k

=dx

ex

1 20 /

... 15

15

2

1

420

tanh

k kk

... 2036

0

ek

kk

!

... log( / ( ))

( )

k k

kk

1

322

... arccosh

... 1

2

2

41

1 2

2 1

1

1

arcsinh( ) ( )!!

( )!!

k

k

k

k

=( )

1 42

1

1

k k

k k

k

.8116826944033783883... 4 2 2 2 12

22

2

log log

H

k kk

k

1 .8116826944033783883...

12

2

22log22log4

k

k

kk

H

1 .8117424252833536436...

1

22

1

4

)4(3)2,2(

jk jkMHS

.8117977862339265120...

H kk

kkk k

( ( ) )log

2 11

11

2

22

... 5

1

( )

!

k

kk

... 6

12

21

4

e

k

kk

k

( )!

... 2

3

e

... 1

2

1

2

1

2

1

22 2 2

2 2 2 ...

=256

3155

1

2

,

.81320407336421530767... 2

3

2

31

1

9 121

sin

kk

J1064

.81327840526189165652...

1

4

2 3

4

3

4 8

1 4

3 4

/

/ sin

1 .81337649239160349961...

... sinh

( )

2

2

4

2 10

!

k

k k GR 1.411.2

= 142 2

1

kk

GR 1.431.2

.81344319810303241352... 11

46 2 4 2

3

42 log log

( )

= ( ) log( )1 12

0

1

x x

xdx

... 4

9

768

18 42

1

log kH kk

k

1 .81379936423421785059... 3

1

3

2

3

hg hg

=d

2 20 cos

= log( )

log( )1

1

1

1

0

3

30

x xdx

x

xdx

=sin

(sin ) /

sin/ x

xE

xdx

1 4 220

2

GR 6.154

... ( )( )

12

k

k

k

k

.8149421467733263011...

2

313

2

3

2

sin k

kk

GR 1.443.5

... 3 2 2 3

3

1

1

4

30

logx

xdx

47 .81557574770022707230... 5

32 1

5 2

20

log x dx

x

=

xx ee

xiLiiLii

4

55

5

)()(2416

5

=x dx

x

4

0 cosh

GR 3.523.7

.81595464948157421514... 3

38

.8160006632992495351...

2

2

1

1

tanh(log )

3 .816262076667919561... 1

2

220 0

0

Ie

I ek

kk

cosh

( !)2

... 6

12

1

2 221

( )

( )f k

kk2 Titchmarsh 1.2.15

.81638547489578... ( ) ( ) ( )3 1 3 1 3 11

k k kk

.8164215090218931437...

( ) ( )

( )

2

2 1

2 1

2 1

1

2

1

212 1

12

12

0

k kk

kk

k kk k

k

... 2

3

1

8

2

0

( )k

kk

k

k J168

.81659478386385079894... 3 2

8

1

1 1

2 loglog

x

x

dx

x GR 4.298.12

1 .81704584026913895975... H

kk

k

( )

!

3

1

... 63

... 33 102

5

1

ek

kk ( )!

... ( )61

5

16

16

1

k

dx

xk

... 3

2

9

4 6

32

21

cos k

kk

GR 1.443.2

= 1

2 23

23Li e Li ei i

4 .81802909469872205712... e

=9

11

... arccsch1

3

... 2 2 14

2 11

1

sin ( )( )

k

k

k

k !

... 3

11

... log

coth

x

xdx2

0

GR 4.371.3

1 .81895849954020784402... ( ) ( )2 3

... 2 6 32 22 2

3

1

log( )

k

kkk

... 1

44 3

1

2 4

2

21

, ,log x

xdx

... 1

11 1e

k

ekk

kk

( ) Berndt 6.14.4

1 .82030105482706493402... ( )

( )!sinh

8 6

2 1

1

1

24

1

k

kk

kk k

... ( )21

kk

... 2 11

2

( )k

k

... x x

edxx

2

0 1

log

... ( )

!

/k

k

e

kk

k

k

3 1

1

1

31

... 2

12

=( )

1 1

21

k

k k J306

=H

kk

kk 21

= kk

k

( ) ( )

1

21

2

=x

edx

x

x xdx

x

xdxx1

1

02

1

2

0

1

log log( )

=x

edxx10

2

log

GR 3.411.5

= log( )log

11

0

1

0

x

xdx e dxx Andrews p.89, GR 4.291.1

=

log( )1 2

0

1 x

xdx GR 4.295.11

=logx

xdx

11

2


=

log x

xdx

10

1

GR 4.231.1

=

log 10

1

e dxx

=log( )

( )

1

1

2

20

x

x xdx GR 4.295.18

= log1

1

2

2 20

x

x

x

xdx GR 4.295.16

= log 11

1

x

dx

x

=log( )

( )

x

x xdx

2

20

1

1

=x

edx

x

3

0

2

1

=e

edx

x

e x

2

0 1 GR 3.333.2

=xe

xdx

x

sin0

= dxx

x

02

2

cosh

... ( )

( )!sinh

4 2

2 1

12

11

k

k kkk

... ( )kk

k 2 12 11

.8235294117647058 = 14

17

.8236806608528793896...

22 2

062

x xd

ex

log x

1 .82378130556207988599... 18 3

22

( )

1 .82389862825293059856... 3

17

.82395921650108226855... 3 3

4

3

21

log sin

x

xk

.82413881935225377438... 1

2

2

0

2

cix

x( ) log

sin/

.82436063535006407342... e

k

k

k

k

kk kkk k2

1

2 2100 0

! ! (2 )!!

.82437753892628282491...

1

22

1)3()3(23

25

k

kk

= k

k kk

1

12 32 ( )

... 1

42 2 1

164 4

1

sin sinh

kk

... )1(4

3)1(

4

32log3

23

4

1

1

khgk

k

k

=

1 )34(

3

k kk

.8248015620896503745...

( )

!

2

1

k

kk

8 .8249778270762876239... 2

1 .82512195293745377856... J J J Jk

k

k

k3 2 1 0

5

20

2 6 2 7 2 21

( ) ( ) ( ) ( )( )

( !)

13 .82561975584874069795... ( )!

12 22

0

ek

k

k

LY 6.41

9 .82569657333515491309... 9

G

2 .82589021611626168568... k

kk !

12

.82606351573817904... ( )1

14 3 22

k

k k k k k

.82659663289696621390... k k k

kkk k

( )

( )

2 1

3

3

3 122 2

1

... sincos ( )

( )!2

1 4 1

1

21 2

2

1 2

2

k k

k k GR 1.412.1

...

2 3 2

2 242

2

2 log

log

= log sin2

0

2x xdx

x

GR 4.424.1

... 3 3

21

1

9 21

kk

GR 1.431

= cos

3 21

kk

= 0

1 3/

.8270569031595942854... ( )33

8

.82789710131633621783... 1

1

1

12

e ii /

... 3 5log

.828427124746190097603... 2 2 21

4 1

2

0

( )

( )

k

kk k

k

k

... 2 2 12 1

2 21

( )

( )!!

k

k kk

!!

... 1

3 320log

x dx

x

...

121

2 14

3

0

Lix dx

ex / 2

... log !

!

k

kk

2

.8287966442343199956... 14 3 161

22

12

12

32

( )( )

( )

kk

= x

e edxx x

2

20 1/

16 .8287966442343199956... 14 31

2

12

12

30

( )( )

( )

kk

... ( )k

kk 21

... 1

21 k kk !log !

... ( ) ( )

( )

( )3 2

1

1 21

k

kk

... i e

e

i i

i i

i

i

1

1

1 1

1

cos sin

(cos sin )

1

... log ( ) kk

2

5 .83095189484530047087... 34 ... H ( )

/2

2 3

... 1

2 2

2 1

122

1

e e

k

e ekkk k

cot( )

8 .8317608663278468548... 78

... 1

31

13

13

( )

( )

k

k pk p

... 2 10

Gx x

edxx

tanh

... 2G

=( !)

( )!( )

k

k k

k

k

2

20

4

2 2 1

Berndt 9

= x

xdx

sin

/

0

2

Adamchik (2)

=x

xdx

cosh0

=arcsin x

xdx2

0

1

1

=arctan x

x xdx

1 20

GR 4.531.11

= GR 6.141.1 K x dx( )0

1

= K x Adamchik (16) dx( )2

0

1

... 3

13

1

9

2

0

( )k

kk

k

k

3 .83222717267891019358... 1

49 3 3

1

1 10

log( )( /

k kk 3)

... 1

43 2

22

0

2

log loge xx dx GR 4.383.1

1 .83259571459404605577... 7

12

... 1 3/

... 45 5

81

0

1 3 ( )log( )

log x

x

xdx

...

log2

... 3

8 2 14 20

dx

x( )

= 5

4

7

4

1

1 4 3 41

k k

k kk

( )

( / )( / ) J1061

.83327188647738995744... Li2

2

3

.83333333333333333333 = 5

6

=( )

1

50

k

kk

=1

1 21 ( )( )(k k kk

3)

=1

4 2

2 2 1

2 21 1k

k kk

k

k

k

k k( )

( )!!

( )! ( )

.8333953224586338242... | ( )|( ) | ( )| k kk

kk

k

kk

1

2 1

2 2

4 11 1

... cos cosh ( )( )

1 1 14

40

!

kk

k k

1 .83377265168027139625... root of ( )x x

1 .83400113670662422217... ( )

( )!sinh

2

2 1

1 1

1 1

k

k kk k

k

... 15 90

90

2 1

4 1

2

4

( )

( )

... 1

3

12

3

2

1

30ee

k

k

k

cos( )

( )!

3 .83482070063335874733... pf k

kk

( )

!

0

... ( ) ( )sin

31

4 32 2

2

31

Li e Li ek

ki i

k

... 2

3 213

8 2 81

2

log/k kk

= 81

2

1

3

( ) ( )

k

kk

k

=( )

( )

k

k kk

1

2

2 121

!

!

... h k

kk

( )2

1

100 AMM 296 (Mar 1993)

.8352422189577681... 1

1 F kkk

... 31 28350 5

84 1

6 6

30

1

( ) log

( )

x x d

x

x

1 .835299163476728902487... 5

24 22 2 2 1 1

12

20

( log ) log( ) logG xx

dx

... 135

453 2 4

4

( )

...

3 3

2

3

1

6 5

1

6 21

log

k kk

J79, K135

=( )

1

3 10

k

k k K Ex. 113

= dx

x x

dx

x

dx

e ex x2 11

30

1

201

... 5

3 327

1

1 3

1

1 3

3 1

3

1

31

( )

( / )

( )

( / )

k k

k k k

1 .83593799333382666716... 1

1 ( !!)!!kk

... ( )k

kLi

k kk k2

22

1

1 1

... ( )

!

1

12

k

k k

.8366854409273997774... e ek e

ek

kk

1 1

0

1

1/ ( )

( )!

6 .8367983046245809349... 28

3 3

32 10

k

k k

k

... 4 1 0

... 4 21

22

0

csclog

/

x dx

x

... Cosh int1

... log( )( )

( )2 1

2

2 11

k

k kk

Wilton

Messenger Math. 52 (1922-1923) 90-93 ... log2

... cos( )

( )!

1

3

1

2 30

k

kk k

AS 4.3.66, LY 6.110

.83800747784594108582... 3

37

...

0

4

)!14()2()2(

1

2

sinh

k

k

kii

=

22

11

k k

... log2

1

2

2

e

e J157

=

1 !

2

k

kk

kk


.83861630300787491228...

1

3

)5(

)3(124

125

k k

k

... G2

... ek

k

k

k

( )( )

!

11

0

GR 1.212

... 5

11

1

1 6

1e

k

k

k

k

( )

( )!

... 531

e ,

... log arctan log3 2 21

21

22

0

1

x

dx

... 2

1 2

2 1 20

1

logarcsin

( )

x

x xdx GR 4.521.6

...

2 21

2

20

2

sin

sin

x

xdx

... 7 45

8

2

7

2

0

e

k k

k

k

!( )

.84062506302375727160... 2 2 1 11

14

40

logx

dx

.840695833076274062...

sin sinh/ /

2 2

21

21 4 1 4

2 42 k

... 23

5

( )

( )

... ( )

1

3 2

1

1

k

k kk

.84100146843744567255... 2

8

... 11

2

kk !

...

22

0

2

24

logcos ( )

sin

/

x x x x

xdx GR 3.789

2 .84109848849173775273...

1

0

1

0

1

0

4

1

log

240

7dzdydx

xyz

xyz

3 .84112255850390000063... H

kk

k

32

1

5 .84144846701247819072...

logcos

sin2 4

210

Gx x

xdx GR 3.791.4

= GR 4.4334.10 log sin10

x dx

... sin Im( )

( )1

1

2 10

ek

ik

k ! AS 4.3.65

= 0

1 /

= 2

1

211J ,

= 112 2

1

kk

= cos1

21k

k

GR 1.439.1

= 14

3

1

32

1

sin kk

GR 1.439.2

= coscoslog

x dxx

xdx

e

0

1

1

... ( )71

6

17

17

1

k

dx

xk

.84178721447693292514... 2 3 11

320

logx

dx

... 1

22 2

2

1 20

Eik k

k

k

( ) log!( )

= 16

19

... 1

4096

1

4

3

4 15 5

5

20

1

( ) ( ) log

( )

x

xdx

... 32

32

1

3

1

1 31 0 10

J Fk k

k

kk

; ;( )

!( )!

...

1 2

1

k kk F

... 1

3 3 331e

B

kk

kk

... erfk k

k

k

( )( )

!( )1

2 1

2 10

AS 7.1.5

= 2 2

0

1

e dxx

... 3 33 1

3 1

11

65

60

log( )

( )( )

k

k k k

... 3 3 4 23

212 42 2 log log log log

= l l

1

6

5

6 Brendt 8.17.8

.8432441918208866651... kk

k

2

1 4 1

... Gx dx

x 2 2

20

1

16

2

4

log arctan

=x dx

x

2

20

4

sin

/

GR 3.837.2

... cos ( )( )

( )!3

0

1 21

3

1

41

1 3

2

k

k

k

k GR 1.412.4

16 .84398368125898806741... e Ik

k kk

20

0

22 1

( )!

... 21

2

1 2

2

1 2

2 1 12

1

1 0

sin( )

( )!

( )

( )!(

k k

k

k k

kk k )k

... 483 2

2

261 2

2

917

4

2log log

=k H

k k kk

kk

4

1 2 1 2( )( )(

3)

... 120 6 360 5 39 4 180 3 31 2 1 ( ) ( ) ( ) ( ) ( )

= k kk

5

2

1 ( )

... Li3

3

4

... cosh( )!

2

4

0 2

k

k k

... Fk

k

k

k 21

=5481948

15625 4

4 2

1

k Fkk

k

... 1

4

1

40 1

1

4 10

, ,( )

!( )

k

k k k

... cos( )

( )!

1 1

20

k

kk k

AS 4.3.66, LY 6.110

... 5

7

1

10

2

0

( )k

kk

k

k

... 9 32

3

1

ek

kk ( )!

... ( )

( )!sinh

2

2 1 2

1 1

22 11 1

k

k kkk k

k

... 22

2

208

1

21 2

1 4

2 2 1

log( ) ( !)

( )!( )

k k

k

k

k k Berndt Ch. 9, 32

= x x

xdx

sin

sin

/

2 20

2

... 3 22

0

x x

edxx

log

=11

13

...

( )

( )

( )6

3 31

k

kk

... 7

621 3

2 1

70

k

kkk

... ( ) ( )

( )! /

1

2

1

22 1

1

k

kk

k

k

k k

e

.84657359027997265471... log log

cosh(log( ))2

2

1

212

23

1

x

xx

dx

x

... 2( ) e

... ( ) ( ) ( ) ( )2 3 2 22

k kk

=2 1

25 41

k

k k kk

3

... ( ) ( )2 3

... ( )

1

15 4 3 20

k

k k k k k k

... sin cos/

x x d0

2

x

... 1

21k

kk F

.8474800638725324646... ( ) ( ) ( )( )

1 1 111

!

e Ei eEiH

kk

k

=

( )

( )!

1

11

k

kk

=1

1

1

111

1

11m k

e

m km

k

k

k

m

k

k!( )!

( )

!( )!

AMM 101,7 p.682

.8474804859040022123... 1 2

31 ( !)k

k

kk

... 4 3

9

1

2 2

12

12

1

( )!( )!

( )!

k k

kk

2 .84778400685138213164... ( )

!

/k

k

e

kk

k

k

4

0

1

41

.84782787797694820792... 7

256

1

4

3

31

k

k

k

sin GR 1.443.5

= 1

4 41 21 4

34

34

i

Li e Li ei i( ) / / /

.84830241699385145616... 3 3 13

313

2

k

k k

kk

( )

.848311877703679...

iLi e Li e

k

ki i

k2

1

222

22

21

( ) ( ) sin/ /

... 8 2

3

32

1

log log( )

x

xdx

...

( )

( )??

( )

( )

k

k

k

k kk k21

12 2

= sum of inverse non-trivial powers of products of distinct primes

... 16

27 1

2

3 20

log

/

x dx

x

.8488263631567751241... 8

31

1

4 22

kk

9 .8488578017961047217... 97

...

eerfi

k

k k

k

k

k

kk

1 40 0

1

2

1

2 1

1

2 1 2/

( ) !

( )!

( )

( )!!

...

1

54 3

1

3

1

3 22

31

( ) ( )

( )

kk

... ( )k

kk 31

... 6

.849581509850335443955... ( )

( !!)

1 1

21

k

k k

...

14

0 )()(

k k

kk

... H kkk

2 11

2 1

( )

=1

2 11

11

11

12

22( )

log log logk

kk

kk kk

.8504060682786322487... 33 1

1 330

e k

k

kk

( )

( )!

... 8 2 12 2

0

1

G x

xdx

log

( )

... sec ( )( )!

1 12

2 k kE

k

... csch12

1

i

i e esin J132

= 21

2 10 e k

k

GR 1.232.3

= 12 2

2

2 12

1

( )

( )!

kk

k

B

k AS 4.5.65

.85149720152646256755...

He k eke

k1

1

11 1

1/ ( )

= ( )k

ekk

12

.85153355273320083373... e e ee k

k

kk

log( )

( )1

1

10

... i i i i

4

1

4

1

4

3

4

3

4

i

=sin

cosh

x

xdx

0

... 7 2log

... 22 1

2

1

22

1 2 1

1 21

log/ ( )

/

( )

( )

k

k kk

Berndt 8.17.8

... 1

43 3

1

2 3

2

21

, ,log x

xdx

... 3 2 4 3 1 32

( ) ( ) ( ) ( ) ( )

k

k

k k k

=2 2

1

4 3

4 22

k k k

k kk

( )

1

... 4

1

2

1

2

11

81 3 2 3 31

( ) ( )/ /

kk

... 4

3

4

32

1

3 21

log( /k kk )

= xe e dxx x

10

GR 3.451.1

.85369546139223027354... 2 24

31

2

2

12

log( ( ) )

k kk

k

=4 1

2 1 22

k

k kk

( )

1 .85386291731316255056... 1

21k

k k

... 5

2

... 1

4

1

42

1

4

1

2

5

41

12

2 1 40

Fdx

x, , ,

= K1

2

... 3 5 5

2

1

5 1

2

0

( )

( )

k

kk k

k

k

... dxx

x

02

161loglog12log24 GR 4.222

... ( )

( )!sinh/k

kk

kk k2 3

1

2

3 2

1

... 2 8

2

4 0

log log cosx x

edxx

... e e

ek

k

e e

k

1

1

121

/cosh sinh(sinh )

sinh

! GR 1.471.1

... bp k

kk k

k

( )

2

1

1 202

0

... 2

1

2

1

2 2 10

erfk k

k

kk

( )

! ( )

6 .85565460040104412494... 47

... ( )

!( )

11

3

2

k

k

k

kk

=k e k k

k e

k

kk

2 1 2

3 12

3 1/

/

... 2 2

52

0

ee x dx

sin x

...

11

11

0

ek k

k

kk

/ ( )

! ( ) )

... ( ) coth24

=6

7

1

60

( )k

kk

...

4G

... coscos1

... ( )k

kk 2 12

.857842886877026832381... 4 53

21

22

0

( ) ( )

x Li x dx

... 3

23 2

1

2 13

2

0

1

( )log

( )( )

Lix

x xdx

2

... 41

2

20

cos

( sin )

x

x

= dxx

x

1

0 1

arccos

= log( )cosh sinh

cosh1 2

20

2

xx x x

x

dx

x

GR 4.376.11

... 35

128 1

7 2

0

1

x dx

x

/

... 24 e

... 2 3

0

2

31

k

k

k

k

k

k

e

k

( )

!

... e Not known to be transcendental

... 2

1

2 1

2

( )

( )!

k

kk

... 96 354

2

1

ek

k kk !( )

... 1

1 231 kk

/

... 4

3

1

31 3

0

ek

k kk

/

!

.86081788192800807778... 4

5

1

2

4

5

1 5

2 21

arcsinh

logF

kk

kk

= ( )

( )

1

2 12 11

22

k

kk

k

k

dx

x x

.8610281005737279228...

2 2

21

4

1

221

coth

kk

J124

.861183557332564183... 1

42 2

1

2 221

( , , )log

x dx

x

.8612202133408014822... ( )81

7

18

18

1

k

dx

xk

.8612854633416616715... 3

36

...

1

0

1

0

1

0

1

0

2

14

3)3(32log8 dzdydxdw

wxyz

zyxw

.86152770679629637239...

11

2

1

20 1

1

2 1

1

1

, ,( )

!( )

k

k k k

.86236065559322214... 1

21k

k k ( )

20 .8625096400513696180... 1

21 2

0ee e

k k

ke e

k

/ cosh

!

.8630462173553427823... 34

3

1

3 1

1

1 30 0

log( )

( ) /

k

kk

xk e

.8632068016894392378... 1 1

21 k

k

kk

log

4 .8634168148322130293... 48 8

22

( )

3 .86370330515627314699... 2 1 312

csc

AS 4.3.46

.86390380998765775594...

1

3

)3(

)3(26

27

k k

k

.86393797973719314058... 11

40

6

60

sin x

xdx GR 3.827.15

.86403519948264378410... 1

83 4 2 42 4

4

1

e e ek

kk

cos coscossin cossinsin

!

... 6 9 12

9

4

9 93

1 31 3 1 3 1 3

// /sec sec sec /


.8645026534612020404... 1

8

3

2

1

4

3

2

3

4

1

2 1 3 20

, ,( )

( ) /

k

k k

.8646647167633873081... 11 22

1

1

ek

k k

k

( )

!

2 .86478897565411604384... 9

.864988826349078353... 1

3 12k

k k( ( ) )

.86525597943226508721... e

1 .8653930155985462163... 7 2 12 4 18 3 1 ( ) ( ) ( )

= ( ) ( ) ( )

1 13

2

k

k

k k k

= 8 3

1

4 3 2

2 42

k k k k

k kk

( )

1

.86558911417184854991... ( )

( )!cosh

8 6

2

11

1

6

14

k

kk

kk k

.86562170856366484625... 1 2

.86576948323965862429... 22

1arctane g d

= arctan(sinh ) ( ) arctan( / )

1 11

1 21

k

k k


.86602540378443864676... = 3

2 3 sin cos

6

=( )

1

12

2

0

k

kk

k

k

... 1

2

5

10 5

1

5 120

csch

( )k

k k

... 1

4 22 1 2

1

4 10

log( )k

k k K135

=1

8 7

1

8 31 k kk

J82, K ex. 113

= 16

1

4 22 2 2 2 2 2

2 2

2 2

arctan arctan log

= 16

4 27

8

3

84 2

8

3

8

cot cot logsin logsin

=dx

x x

dx

x2 21

40

1

1

... 2

36 2

1

1 1 40

( log )( )( /

k kk )

... 21

2

1 22

1

21

arcsinh

( )k k

kk

k

k

5 .8673346208932640333... 1

128

1

4

3

463 3

3

0

( ) ( ) tanh

x x

edxx

11 .8673346208932640333... 1

128

1

4

3

43 3

3

0

( ) ( )

cosh

x dx

x

5 .86768692676152924896...

1

)(

k kF

k

...

1 2 ),2(

)1(

k

k

kkS

1 .86800216804463044688...

2 13 4 40

/ /

dx

x 2

.86815220783748476168...

13 3

108

1

432

1

6

5

62 2

( ) ( ) ( )

= 3 30

1

3 1

1

3 2

( )

( )

( )

( )

k k

k k k 3 J312

.86872356982626095207... ( )31

3

1 .86883374092715470879... )2,4(2268

)3(6

2 MHS

.86887665265855499815... ( )

( )log

1

2 11

1

21 1

k

kk

kkk

... ( )

( )!cosh

2

2

11

2 1

k

k kk k

... 1

22

22 2

1

e ek

ke e

k

k

/ cos

!

.869602181868503186151... 2

16

1

2

1

2

1

2

1

2

2 2

1 1

( ) ( )

4

2 2

16 2cot

= H

kk

k 4 221

... 2 2

0

1

1

logsin

sin

x

x

dx

x GR 4.324.1

... 21

1

2

1

3k

k

k

k

k

ke

( )

!/

.87005772672831550673... 4

2

2

2

0

log

loge xx dx GR 4.333

3 .87022215697339633082... I Ik

k

k

k

k

kk k0 1

3

21

3

0

2 22

2( ) ( )

( !) ( )!

.87041975136710319747... 21

32

1

2arcsin arctan

=( )

( )

( )!!

( )!!

1

2 2 1

2

2 1 30 1

k

kk

kkk

k

k k

...

4

23

1724 2 2 3

1

2 1

log ( )( )

( )

H k

k kk

k

... 1 4/

... 23 e

... 2

21

28 21

1

4 1

1

4 1

( )

( )

( )

( )

k

k

k

k k J327

= 1

64

9

8

7

8

5

8

3

8

=

log x dx

x40 1

... 5

16

1

16 2

3

1

e

e

k

kk

( )!

...

( )

( )

k

kk

1

12

... 29

207 1 ( )

... x dx

e ex x

2

1

... 15

... ( ) ( ) ( ) ( )

1 3 1 3 3 11

1

k

k

k k k 3

... 4 82)

3

1

ek

k kk

!(

... 4 11

4

1

ek

kk

( )!

... 4e

... 11

231

k

.87298334620741688518...

k

k

kkk

k 2

)1(6

)1(315

0

1 .8732028500772299332... 11 2

2 2

( )

( )

( )

( )

k

k

k

kk k

1

.87356852683023186835... 1

log J153

1 .87391454266861876559... 11

21

k

kk

... 3 2 2

20

2

20

2

8 2 2 1 2

log log logx

xdx

x

xdx

7 .8740078740118110197... 62

... 1

6 2

1

42 3 2

0

23 2

0

2

sin cos/

//

/

x dx xdx GR 3.621.2

... ( )

( )!cosh

k

k k kk k2 2

1 1

2 1

k

1

... ( ) ( )4 2 4 20

k kk

... 1

43

3

23

3 3

3 3 32

1

log log

H kk

k

.87446436840494486669...

( ) ( )

( )

int

k kk

nkk

a nontrivialeger power

nk

11

2 22

... 3G

= 7

8

1

72 1 2 1 1

0 1

( )( ) ( )

k

kk k

k k

... 3 2

082

x x

edxx

tanh

... 3 2

20

2

08 1

log

cosh

x

xdx

x

xdx GR 3.523.5

= 3

3 3

2

42

i Li i Li ix

e ex x( ) ( )

... cos( )

( )!

1

2

1

2 40

k

kk k

AS 4.3.66, LY 6.110

... G

... 2

21

2

1

21

log( )k

k k k

= log 1 2

0

1

xdx

x GR 4.295.2

= log( )

11

1 20

xdx

= log1

1

2

20

1

x

xdx

=x dx

x10

2

cos

/

= dydxyx

yx

1

0

1

0221

... coth

... 109

81 3

1 3 4

1

e k

k kk

/

!

5 .8782379806992663077... 3

82 4

1

2 2

3

21

, ,log x

xdx

... 2 21

24

1

24

1

2 2 10

sin cos( )

( )! ( )

k

kk k k

.879103174146665828812... ( )( )

log

12 1 1

111

12

1

k

k k

k

k k

k

... logk

kk2

1 1

...

1 )1(!

1

k kkk Related to Gram’s series

.87919980032218190636... 2

2 10

k

k kk

( )

.87985386217448944091... k

k kk

!

1

.8799519802963893219...

0

1 113 1

1

3 1

( )kk

kj k

jk

= 1

3111ijk

kji

.88010117148986703192... 2 11

1 410

Jk k

k

kk

( )

!( )!

1 .880770870194...

0

1 11

11

1( )

( )

k

k k k kk k

=1

111 jk jkkj ( )

... I0 3( )

... e

e e

k

kk

2 201

1 1

2

1

tanh ( ) J994

5 .88097711846511480802... 2 1

2

1

2 21

2

1

4 2F

k

kk

, , ,( !)

( )

2 !

... 0 1

2

0

41

26

1

3 2F

k k

k

kk

; ,( )

!( )!

... 9 3 12 2 3 31

1 5 60

log log( )( /

k kk )

= ( )k

kk 6 2

2

... cosh2

2

...

tanh/2

1

1 220

k kk

=

2

1

2

1 iii

... log(1 + 2) = arcsinh11

4 2 1

2

0

( )

( )

k

kk k

k

k J85

= log tan8

= arctanh1

2 i iarcsin

.88195553680044057157...

k

kk

12

21 2 2 1( )

1 .88203974588235075456... 1

121 k kk log( )

... 2

21 2

2 1

2 1

0

erfk k

k k

k

( )

!( )

=15

17

... e k

k

1

2

1/ !

...

1

3

3

63

1

322

csc

( )k

k k

... 2

4 11

k

kk

3 .88322207745093315469... 5 1 14

120

logx

dx

.88350690717842253363... ( ) ( )2 1 2 11

k kk

3 .8836640437859815948... e EiH

kk

k

1 111

( )( )!

... HypPFQk k

k

k

11 2 2 2 11 1

21

, , , , ,( )

( !)

... 4

81 3

3

3

g J310

1 .88410387938990024135... 64

10395

5volume of the unit sphere in R5

... e log 2

1 .8841765026477007091... 11

4

3

27

0

2

x e x dxx log

.88418748809595071501... H k k

kk

k

! !

( )!21

=

2

13

2

31

3

3log

3

2

33

222

iLii

iiLi

... 16

17

16

17 17

1

41

2 40

arcsinh ( )! !

( )!k

kk

k k

k

3 .88473079679243020332... 22

2

2

2

2

1

GH k

kk

k

k

log ( !)

( )!

... 2

31 3 3 3

3

2 10

cosh sinh( )

k

kk k

1 .88491174043658839554... Gx dx

x

2 2

2 20

1

32 1

log

( )

1 .8849555921538759431... 3

5

... 2 2

2 2212

1

16 1

k

kk ( )

= k kk

( )2 11

... e EikH

k

kk

k

2

1

1 2 2 2 2 12

( ) log!

... 612

4 2 22

2 2

0

1

log log arccos logx x dx

... 2 2 3 ( ) ( )

=2

1 21

H

k kk

k ( )

=

log

( )

2

21

2

20

2

01 1 2

x

x xdx

x dx

e

x dx

e e ex x x x

... 8

1 .88580651860617413628...

2 4

1)(5

8

5

4

2log15

4

5

kk

k k

... 3

35

5 .8860710587430771455... 16

4 1e

( , )

... 5

.88620734825952123389... 2

31 3

2 1

1 2 1

0

erfk k

k k

k

( )

!( )

.88622692545275801365... 2

AS 6.4.2

=( ) ( )!

!

1 31 12

1

k k

k

k

k

= 3

2

AS 6.1.9

= sin2 2

20

x

xdx

= e d LY 6.29 xx

2

0

= ex

xdxxtan sin

cos

2

20

GR 3.963.1

... 24 5 11

4

0

( )

x

e edxx x

... 24 5 11

44

0

1

( )log( )

x

xdx

... 2 21

2log

... 5

2

2

22

2

22 2 12

2 1

cot ( )

kk

k

k

k

... ( )

11

1

k

kk k

... 5

3

... 1

2

2

3 3

1

3 120

csch ( )k

k k

... sech1

2

2

2 41 2 1 22

0

e e

E

kk

kk

/ / ( )! AS 4.5.66

...

log log

1 3

2

1 3

2

i i

= log ( )( )

11

13

31

1

1

k

k

kk

k

k

... log

( )

4

2 1

k

k kk

... 9 3log

... ( )

1

3 2

1

1

k

kk

... F

kk

k

2

1 !

.8876841582354967259... 2 21

2

1

1 2 10

log ( )( )

( )! (Ei

k k

k

kk )

... k kk

5 2

2

1 ( )

... 3 3

6

CFG D14

8 .8881944173155888501... 79

... ( )

1

5 10

k

k k

=8

9

1

80

( )k

kk

... 253

5

1

1 5

1

1 52

2 1

2

1

21

( )

( / )

( )

( / )

k k

k k k

... 1

1 11 k kk ! ( )

... 11

2 41

k k kk ( )( )

...

41

2 13

2

0

Lix dx

ex / 2

... 2 3 3

81 13 20

log

( )

x

xdx

... 15

8

...

2 2

3

23

5

6log log

... 1

42 3

1

2 2

2

21

, ,log x

xdx

.890898718140339304740... 2 11

6 11 6

1

/ ( )k

k k

.8912127981113023761... HypPFQx dx

ex[{ , , },{ , , }, ]log

111 2 2 2 11

2

2

0

1

=( )

!( )

1

1 30

k

k k k

... 3

8

5

2 1

4

0

2

x dx

ex

...

41 2 2

0

2

e dcos (tan)/

x GR 3.716.10

= dxx

x

02

2

1

cos

.891809...

1 )1(

)(

k kk

k

... dx

x x21 1log( )

... 45

4

1

4 1 10 0

log( )

( ) /

k

kk

xk

dx

e 4

... 1 3

1 2

/

/

... 3 23log

...

0

2/

2! k

ke

k

ee

... 11

2 kpp

p primep primek

( )

... ii eLieLi 24

24

... 4

3

1

3

1

3

3

0

e dxx

... HypPFQk

k

k

13

2

3

2

1

4

1

2 1 20

, , ,( )

(( )!!)

=

210

0

2

H x( ) sin(cos )/

dx

... x x

edxx

2

0

1

1

log( )

...

1

2

)2(4

360

7

k

k

k

H

= dxx

xx

1

0

2log)1log(

... 22

2cos ( )/ ( e i ii ii i

)

... 82

1

96

1

414 3

1 42

33

2

41

Gk

kk

( ) ( )( / )

.8944271909999158786... 2

5

1

16

2

0

( )k

kk

k

k

=17

19

.89493406684822643647...

2

22 26

3

4

2 1

11 1

k

k kk k

k

k

k( )( ) ( ( ) )

= 1 122 3 2

2 2k k k kk k

k k

( ) ( )

= 2

2

1

1 13 31

2

22k k

k k

k k kk k

( ) ( )

1 .89493406684822643647... 2 2

0

1

6

1

4

1

1

( ) logx x x

xdx

.895105505200094223399... ( )

( )!

/k

k

e

kk

k

k

1 1

11

1

22

1 .89511781635593675547... Ei li ek kk

( ) ( )!

11

1

AS 5.1.10

.89533362517016905663... si

k k

k k

k

( ) ( )

( )!(

2

2

1 2

2 1 2 10

)

.89580523867937996258... iLi e Li e

k

ki i

k2 4 41

( ) ( )sin

4

... 2

4 2

1

2csc

= ( )

1

2 1

1

21

k

k k

.89601893592680657945... sin

2 21

1

16 121

kk

J1064

.89682841384729240489...

2

1

2

1

22 12Li , ,

= log log log log2 222 2 2 3 3 2

1

3

Li

=( )

( )

1

2 1 20

120

k

kk

xk

x

e

... ( )

!

/k

k

e

kkk

k

k

3

20

1 2

31

... 2

11

... 7 3 1

6 2 1 12 70

dx

x /

=115

128

1

53 0

5

3

1

, ,k

kk

... 1

2 1

2

1

log

( )

k

k kk

.8989794855663561964... 2 6 41

8 1

2

0

( )

( )

k

kk k

k

k

4 .8989794855663561964... 24 2 6

.89917234483108405356... 3

2

1

3

1

3 2 10

erf

k k

k

kk

( )

! ( )

.89918040670820836708... 22 2 2

1

2 121

cot

/kk 4

.89920527550843900193... 3

4 17

17

2

1

3 221

tan

k kk

9 .8994949366116653416... 98

... 1

1 G

.90000000000000000000 = 9

10

1

90

( )k

kk

.9003163161571060696... 2 2

11

16 21

kk

GR 1.431

= 0

1 4/

= cos

2 21

kk

GR 1.439.2

2 .900562693409322466279... 3 3

16 24 22

22

0

1 ( )log log arccot x x dx

1 .90071498596850182787... 2

1

2

2

( )kk

k

2 .90102238777699398997... 1

1 21 kk ! /

... 11 11

5

1

ek

kk

( !

...

0

2

32

)1(2

2

12

kk

k

k

kK

... 1

2

2

42

0

1

si

x x d( )

log cos x

... 3 3

43 1 3 0

1 1

31

( )( ) , ,

( )

k

k k AS 23.2.19

= log( )log

10

1

xx

xdx GR 4.315.1

= x dx

ex

5

0

2

1

.901644258527509671814... 2 10

11

3

4

3

1

2

1

2 3 1F

k

k

kk

, , ,( )

( )

... Brun’s constant, the sum of the reciprocals of the twin primes: primeatwinp p

1

Proven by Brun in 1919 to converge even though it is still unknown whether the number of non-zero terms is finite.

... ( )( )

11

21

2

1

kk

k

k k

... cos 2

... 5

3

... ( ) ( )

1 13

2

k

k

k

... sin Ime

ie2

...

( )

( )

k

kk

3

12

... 6

3

62 3

1

6 10

log( )k

k k

... 1

3 10k

k

1 .9041487926756249296...

3

2

3

2

3

2

3

4,2,

2

3F

3

210

3/211 IIe

=

k

k

k

k

kk

2

3!0

= 6463

512

1

55 0

5

5

1

, ,k

kk

... 3

82

7 3

4

16 6 1

2 2 1

2 2

3

log( ) k k

k kk

=( ) ( )

1

2

2

2

k

kk

k k

... ( )

( )!( )

1

2 4 10

k

k k k J974

= 2

2 12

0

1

21

2C x dxx

dx

x

cos cos

... e

e

3

3

1

1

J148

... 4108

243

1

45 0

4

5

1

, ,k

kk

... ( )k

Fkk

1

12

... e

.90575727880447613109... si

k k k

k

k

( ) ( )

!( )( )

2 1

2 1120

... 2 2 22 1

11

1 22

1

1

log log( )

( )

( )(

kH

k

H

k kk

kk

kk

)

=

12 24k

k

kk

H

... 1

2 2 11k

k k (

... 3

53 3 3 4 2

1

1 1 60

log log( )( /k kk )

... e

3

... 5

4

1

4

1

4

4

0

e dxx

... 1

1 k kk !

... 2 1

21

2

1

22 1

1

16 1

2F

k

k

k

k

k, , ,( )

( )

.9066882461958017498...

2 0

log sinx x

xdx

.9068996821171089253...

2 3

1

6 1

1

6 50

k kk

J84, K ex. 109e

= maximum packing density of disks CFG D10

= ( )

( )(

1

1 3 10

k

k k k )

= ( )

( )

1

3 2 10

k

kk k

J273

=dx

x

dx

x x

dx

x20

4 20

203 1

3 1

= x dx

x x

2

4 20 1

= xx

dx2

0

1

311

log

1 .90709580309488459886...

1

2)12(

31

2

3cosh

4

1

k k

.90749909976283678172... b k

k

k

kk

k k

( )( )

( )log ( )2

1 1

2 2

Titchmarsh 1.6.3

.90751894316228530929... ( ) logk kk

12

23 .90778787385011353615... 5

64

5 4

0

x

e edxx x

= GR 4.227.3 log tan/

x dx4

0

2

=log4

20

1

1

x

xdx

GR 4.263.2

... Lii

Lii

2 2

1

2

1

2

.90812931549667023319... 4 2

4190 48 96

1

768

2

cos( / )k

kk

GR 1.443.6

= 1

2 42

42Li e Li ei i/ /

.90819273744789092436... 1

32

5 3

2

5 3

2

i i

= k

kk k

k k

1

13 1 3 23

2 1

( ) ( )

.90857672552682615638... ( ) ( )

( / )1 2

31

4

3

1

6

4

3 1

k

kk 3

.90861473697907307078... 0

1 1 112

1 2 1

2

( ) log( )k

k kkk

k

kjk

kj

jk

... ( ) ( )

1 2 22

2

k

k

= 10

11

1

100

( )k

kk

... 1

2 k kk !log

... sin( )

( )!

sin( / )

!2

1 2

2 1

2 22 1

0 1

k k

k

k

kk

k

k

AS 4.3.65

... 1 1 1

2 1

ex dx

e(cos sin )sinlog

... 3 3 2 2 12

0

1

log log log

x

dx

... 5

23 5

1

5 1

2 2

0

kk k

k

k( )

... 6

1 .91002687845029058832... 1 2 35

44 1

31

( ) ( ) ( )H

kk

k

.91011092585744407479... ( )2

2

1

221

2 21

k

kLi

kkk k

.91023922662683739361... 1

3 30log

dx

x J153

... 1

8 42

1

420

cothkk

.91059849921261470706... sin(log ) Im i

.91060585540584174737... Li3

4

5

1 .910686134642447997691... 1

2 2

1

2

2

21

eerf

k k

k

k

k

!

( )!

244 .91078944598913500831... 8

396 64 2

33 2

22

0

log ( )/x Li x dx

.911324776360696926457...

2

1)()(k

kk

.91189065278103994299... 9

2

.91194931412646529928... 3

34

23 .9121636761437509037... 65

5 1e

( , )

... 1

11219 14 2 2 2 csc( )

... 73

...

121 k

kk

...

( )32

... 5

22 CGF D4

.9142425426232080819... 7

2

7

1

7 220

arctandx

x x

=32

354 1 2

1

4 40

( , / )( )k

k k

2 .91457744017592816073... cosh( )

33

20

k

k k !

... 2

1cot

2

1

=

i e

e

i i

i

B

k

i

i

kk

k

1

2 1 2

1 1 1

1 1 1

1

22

0

cos sin

cos sin

( )

( )!

= limsin

aa

k

k

k

0 1

.91547952683760158139... ( )

1

7 10

k

k k

... 120 46 1255

4

1

( )!( )

ek

k kk

...

32 3

1

1 320

tanh/

k kk

.91596559417721901505... G ( ) Im ( )2 2Li i , Catalan’s constant LY 6.75

=

2

1

2

1

28

2log22

iLi

iLi

i

= ( )

( )

1

2 1 20

k

k k

= 1

4 3

1

4 12 21 ( ) ( )k kk

= 1

2

4

2 2 1

2

20

( !)

( )!( )

k

k k

k

k

Adamchik (23)

= 1

16

3 1 1

42

1

( )( )( )

k

kk

kk

Adamchik (27)

= 1

8 21

3

41

kkk

k

, Adamchik (28)

= 8

2 33

8

12

2 1 20

log

( )

k

kk

k

= x

e edxx x

0

=

log logx

xdx

x

xdx2

0

1

211 1

= log1

1 10

1

2

x

x

dx

x GR 4.297.4

=

log

1

120

1 x

x

dx

x Adamchik (12)

=

log1

2 1

2

20

1 x dx

x Adamchik (13)

= GR 4.227.4 log tan/

x dx0

4

= 1

2 0

2 x

xdx

sin

/

Adamchik (2)

= 1

2 0

x x dsech

x Adamchik (4)

= ar Adamchik (18) csinh(sin )/

x dx0

2

= Adamchik (5), (6) 2 2 2 20

4

0

4

log( sin ) log( cos )/ /

x dx x d

= arctan arctanx

xdx

x

xdx

0

1

1

GR 4.531.1

=

arctan x

xdx

1 20

= 1

22

0

1

K x dx( ) Adamchik (16)

= 1

22

0

1

E x dx( ) Adamchik (17)

= 1

1 10

1

0

1

( )x y x ydx dy

Adamchik (18)

.9159981926890739016... ( log ) log( )

( )

8 3 18 2

12 2

1

1

4

4 40

x

x xdx

5 .91607978309961604256... 35

... Li13

5

.91666666666666666666 = 11

12

1

110

( )k

kk

... cos cosh( )

( )!2 2

1 2

44 4

0

k k

k k

... HypPFQk

k

k

k

k

, , , ,( )

( )!

1

4

3

41

1

64

1

4

2

0

... 2 4 ( )

.91775980474025243402...

93 3 1 3

1

1

3

3 30

(log )log( )

( )

x

x xdx

... 6

5

... 21

2 2

1

2

1

2 1 21 4 1 20

sin( )

( )!/ /

J

k

k

kk

... 7

16

13

32

1

2 21 4

2

0

e erfk k

kk

/ !

( )

!

... log lim!

/2 1 2

n

n

n

n e

n

= Stirling’s constant , GKP 9.99

the constant in ...12

1

2

loglog!log

n

nnnnn

= GR 6.441.2 log ( ) x dx0

1

=1

1 20

1

log logx

x

x

x dx

x x

GR 4.283.3

.9190194775937444302... sinh

exp( )

41

1 44

2 1

k

k

kk

1

= 1

2 2 2 i i

... 7

4

... ( ) ( )( ) ( )1 1i i

... H

kk

k

2

1 !

.9193953882637205652... 2 2 1 41

2

1

2 12

1

cos sin( )

( )!

k

k k k

... 12

2 1

2

20

sin

sin

x

xdx

.920673594207792318945... e

e e

k

ekk( )

, ,

1

11 02

0

=( )k

ekk

11

... ( ) !

( )! /

1

2 1 2

2 1

2 201 8

k

kk

k

k eerfi

... log log( )2

22

1

1 20

1

G

x

xdx GR 4.292.1

= arccosx

xdx

10

1

GR 4.521.2

= x x

xdx

sin

cos

/

10

2

GR 3.791.9

12 .92111120529372434463... 2 82

1

1 420

csch ( )

/

k

k k

... 2

2 2

2 2 2 2

sin sinh

cosh cos

= 11

11 1 44

2

1

1

k

kk

k

k

( ) ( ) 1

... 4 21 2

2

1

log( /

H

k kk

k )

7 .9219918406294940337... e I

k

k

kk

20

0

2

2

1

2

1

1

2 1( )

( )!

... 2

216 4

1

16

2

cos( / )k

kk

= 1

2 22

22Li e Li ei i/ /

... erfk kk

k

1

2

1

4 2 1

1

0

( )

! ( )

... ( )x d 10

1

x

... 3

23 ( )

... 2

1

2 1( )kk

=12

13

1

120

( )k

kk

... ( ) ( )

( )!sin

1 4

2 1

1 11

2 211

k

kk

k

k k

k

...

4 1

4 1

2

12

0

tanhk kk

.923879532511286756128... 8

3sin

8cos

4

22

... 90 1

44 41

( )

( )k

kk

... point at which Dawson’s integral, , attains its e e dx t

x

2 2

0

t

maximum of 0.5410442246. AS 7.1.17

.92419624074659374589... 4 2

31 2

1

27 331

log ( )

k

k k k Berndt 2.5.4

= Hk

kk

/2

1 2

... ( )

log

1

2

k

k k

... 2

5025 5 5 1

cot csclog

x

xdx

... ( )

1

8 10

k

k k

5 .924696858532122437317...

2 3

1)(5

36

5

2

3log5

3

5

kk

k k

...

1

32

22

1

22

)14(

)14(41)2(

432

3

kkk k

kkk

k

= dxx

x

121

arctan

... erfik

k kk

( )( )

!( )!1 2

12

121

!

... sec1

2

1

e ei i

... 1

4 1

1

23 1

1

21

1

2( )( ) cosh ( ) sinh

ee e

sech

=( )

1

12 20

k

k k

... 11 12

2 21

sinh ( ) ( )k i k ik

2

=2 2

2 1

2

4 21

k

k kk

... 3 2 5 ( )

... 12

2

cosh

... 2

2128

22

4 1

2 2 1

log( )

( )

k k k

k kkkk

... 1

122

2 12 2

3

0

loglog

( )( )

x

x xdx GR 4.262.3

... 1

8

1

4

1

4

1

2

5

41

12

2 1 41

Fdx

x, , ,

.927099379463425416951... 3 3

26 2 3 6 2 3

1

4

22

2

loglog log log

Li

=( )

( )

1

3 1 20

130

k

kk

xk

x

e

...

1

5 3 5

10

5 5

2

5 3 5

10

5 5

2

= 2 1

11 12

21

2

k

k k kF k

k

kk

k

( )

( ) ( )

... 21

21

21

2

1

4 2 10

arctan log log( )

( )

ii

ii

k

k

kk

... 7

4

2

2 ( )

... 1

2

23

23

23

1

Li e Li ek

ki i

k

/ / cos( / )

...

4

41105

8

4

( )

( )

( )k

kk

HW Thm. 300

2 .92795159370697612701... e2 1 2 1 2 1 2 1 2 1cos sin( sin ) sin( sin ) cosh( cos ) sinh( cos )

=

ie e

k

ke e

k

k

i i

2

22 2

1

sin

!

1 .92801312657238221592... 2 1 2 14

20

( log ) log log

xx dx GR 4.222.3

3 .928082040192025996... pd k

kk

( )

!

1

.92820323027550917411... 4 3 61

12 1

2

0

( )

( )

k

kk k

k

k

6 .92820323027550917411... 48 4 32

61

2

k

k

k

kk

13

14

1

130

( )k

kk

... 2 3

0

1

3

49

36

1

1

( ) logx x

xdx

... 1

22 2

2

220

Ik k

k

k

( )!( )!

... 5

2 2

1

1

55

51

i e

e

k

k

i

ik

logsin

... e

... ( ) ( ) ( )2 3 4 312

42

k k

kk

... ( ) ( ) ( )2 3 412

41

k k

kk

... ( )

!/3

11

1

1

3k

ke

k

k

k

... 3 3 1 19

20

log log log

xx

dx GR 4.222.3

... 2 10

11

3

4

3

1

3

1

3 3 1F

k

k

kk

, , ,( )

( )

... 263

2310 2 1515

csc

... e

... ( )

!( )!

k

k k kI

k kk k

1

12

1 1

21

1

.93122985945271217726... 21

2

1

8 2 1

2

0

arcsinh

( )

( )

k

kk k

k

k

.93137635638313358185... | ( )| k

kk 21

6 .93147180559945309417... 10 2log

.9314760947318097375...

( )( )

3360 1

4

31

k H

kk

k

1 .9316012105732472119... e e k

k e

e

kk

2

2 0

cosh

!

... 3 4

32 2 14 20

log

( )

x

xdx

... ( )

1

9 10

k

k k

...

3

2 3 21

log/

k

kk

.9326813147863510178... 4

21

1

162 41

sinh

kk

.932831259045789401392...

1

02

22

123

12

)(2

)1(2log2

12 x

dxxLi

kkk

k

1 .932910130264518932837... 37

163

12

2

22

0

( ) ( )x Li x dx

... cos( )

( )!

1 1

2 20e k

k

kk

e

AS 4.3.66, LY 6.110

... 8777

1

ek

kk

!

=14

15

1

140

( )k

kk

5 .9336673104466320167... 1

1256

1

4

3

4 13 3

3

21

( ) ( ) log

x dx

x

...

1

1

... 11 1

1

ee

ke

k k

k

( )

!

... 413 1 10

0e

k

k

k

k

( )

!

.93478807021696950745... 2 7

6

4

3

12

13

161

k k

k kk

( )

( )( ) J1061

.93480220054467930941...

2

3

)12(!)!12(

!)!2(4

2)1(

12

2

k kkk

k

=1

1 2

1 1

2

4

2 121

22

21( / )

( ) ( ) ( )

( )k

k k

kk

k

kk k

1 .93480220054467930941... dzdydxxyz

zyx

1

0

1

0

1

0

2

13

2

2 .934802200544679309417... 2 2

022

1

x x d

x

sin

cos

x

4 .93480220054467930941...

2

1

2

1

2

( ) logi i

= 3 21

21

212

20

( )( )

,

kk

=

211 212 )!2(

4)!1()!1(2

4

!)!12(

!)!2(

k

k

k

k

k k

kk

kk

kkk

k

= k k kk k

k kk k

2

2

2

2 22

14 1

1 ( ) ( )

( )

= 2 12arcsin

= log( )

2

02

1

1 1

x

x

dx

x x GR 4.297.5

= logtan

tan2

0

1

1

x

dx

x GR 4.323.3

8 .93480220054467930941...

2

1

24

1

2

( )

... 7

7291

1

3 1

1

3 1

4

4 41

4

( )

( )

( )

( )

k k

k k k

... 5

2 25 5

0

1 51

1

25 21

/ kk

... ii eLieLi 2

32

3

... log( )

( )(2

3 2

1

2 1 3 10

k

k k k )

... 11

12

( ) ( )k kk

.936342294367604449... sin1

11

12 2

2

kk2

... x dx

e ex x

3

1

.93709560427462468743...

81 2 2 2

0

2

( log ) log(sin )cos/

x x dx GR 4.384.10

= 1 2

0

1

x x dlog x GR 4.241.9

= xe e dxx x

1 2

0

GR 3.431.2

7 .9372539331937717715... 63 3 7

... 3

2 2 6

2

2 11

2 2

1

log( )

k

kk

k

Admchick-Srivastava 2.23

=15

16

1

150

( )k

kk

... ( )

1 1

1

k

kk k

k

...

( )log

2 21

k

kk

Berndt 8.23.8

= 1

1 311 ( )!

log

n

k

k

n

kn

... ( )

1 1

12

k

kk k

1 .93789229251873876097... 3 2

2 20

2

2016 1

log

( )

x dx

x

x dx

e ex x

= i Li i Li i3 3( ) ( )

= log log2

20

1 2

211 1

x dx

x

x dx

x

GR 4.261.6

=

log2

40 1

x dx

x

= GR 4.227.7 log tan/

x dx2

0

4

= dydxyx

yx

1

0

1

022

22

1

)log(

.93795458450141751816... )()(31524

1

1222

24

iiik

H

k

k

.93809428703288482665... ( )

1

10 10

k

k k

... 81

24

1

28

1

2 4 10

cos sin( )

( )! ( )

k

kk k k

... Jk

k

kk

1 20

1

2

1

16

( )

( !)

... ( ) ( )

( )! /

1 2

1

11

12 1

12

k

kk

k

k

k k e

... k Hk

kk

2 3

1 2

( )

... 12

1 1( ) ( )k kk

...

2

coth

... 5 5

4

5 5

5 5

25 5

4

5

21

2

5log

log

=5

5 1

1

1 4 5 51 02

2k k k k

k

k kk

k( ) ( )( / )

( )

4 .939030025676363443... 3 24

256 2 23040

12 4

41

( )

( )

( )

a kk


... 3

33

1 .9395976608404063362... log log21

2

2 2

2 2 22

1

H kk

k

4 .9395976608404063362... 3 21

2

2 2

2 2 22 1

1

log logkH k

kk

...

primekthp

k

k

pk 1)()(

...

primep pp

p42

22

1

1

)4(

)6(

21

2

... J310 g4

.94015085153961392796... ( )

( )

k

k kk

12 !

... ( )

!

/2 1

1

1

1

2

k

k

e

k

1

k

k

k

... 115

384

5

50

sin x

x GR 3.827.11

... 24 2 321

8

2 2

0

kk

k

k

=16

17

1

160

( )k

kk

... 89

8

1

8 10

log( )

( )

k

kk k

... 42 2

3 CFG D4

.94286923678411146019... 2

316 4

1

12

sin k

kk

Davis 3.32

= iLi e Li ei i

2 3 3

.942989417989417 = 7129

7560

1

3 31

k kk ( / )

.94308256800936130684... HypPFQ[{ , , , },{ , , , }, ]1 1 1 1 2 2 2 2 1

=( )

!( )

1

1 40

k

k k k

... 5

4

2

42

1

222

cot

kk

= 2 21

1

k

k

k

( ) 1

... 3 2 66

1

1 620

csch ( )

/

k

k k

.94409328404076973180... ( ) arcsin arcsin3

4

2 3

0

1

x x

xdx

.9442719099991587856... 4 5 81

16 1

2

0

( )

( )

k

kk k

k

k

8 .9442719099991587856... 80 4 5

=17

18

1

170

( )k

kk

... cos( )

( )!

1

3

1

2 9

k

kk AS 4.3.66, LY 6.110

...

21

2

1

2 13 30

Lik

k

kk

( )

( )

=log2

21

2

02 2

x

x xdx

x

edxx

1

= log2

0

1

2

x

xdx

.94529965343349825190... 0

1 1

( )

( )

k

kk

!

.94530872048294188123... 8

15

5

2

1 .9455890776221095451... ( )

( !)

2 212

10

1

k

kI

kk k

... log76

71

Li

.94608307036718301494... 1

0

sin)1( dx

x

xsi

=( )

( )!(

1

2 1 2 10

k

k k k ) AS 5.2.14

= sin1

1 x

dx

x

= 1

0

coslog dxxx

... 320

epf k k

kk

( )

... sink

kk 21

...

52 5

1

1 520

tan/

k kk

... 1

43 1 31 3

3

1

e ek

kk

cos cossin(sin ) sin(sin )sin

!

= ie e e ee e e ei i i

83 3

3 3

i

... 7

7204

7 4

8

14 1

41

( )( ) ( )k

k k J306

.947123885654706166761...

2

1

42

1

23 41 4

1 4//

/cot ( ) coth( )

i i

= k

kk

2

41 2

.94716928635947485958... 11

11

2

1

321

tanh

k kk

... 18

19

1

180

( )k

kk

...

162 8 2 82 2 2 log log 2

dx

= GR 4.335.2e xx

2 2

0

log

...

sin1

113 2

1

kk

... 8

1 21

8 7

1

8 11

k kk

J78, J264

... Q k

kk

( )

!

1

... ( ) ( )2 3

3

4 .949100893673262820934... 1

3 1 ( )

.94939747687277389162... arctan(sinh csc ) ( ) arctan1 1 11

11

k

k k


.9494332401401025766... k k

kk

kk3 1 31

1

1

( )

... 2

6

.94949832897257497482... 2 1 1

4120

sin ( )

!( )

k

kk k k

... 2

2 216 3

11

6 1

1

6 1

( )

( )

( )

( )

k k

k k k J329

=

log x

xdx6

0 1

9 .9498743710661995473... 99

=19

20

1

190

( )k

kk

... 2

22

206

31 5

2

1

2 2 1

log( ) ( !)

( )!( )

k

k

k

k k

...

6

3

2

1

3

1

3 221

coth/

kk

.95075128294937996421...

1

3

)2(

)3(7

8

k k

k

.95105651629515357212... sin2

5

.9511130618309... ( )( )

1 14

1

k

k

k

k

1 .95136312812584743609... 2 21 8

22

1

1

sin( )

( )!

k k

k k

... 15

8

2

6

2

0

e

k k

k

k !( )

... 1

36

1

6

2

3

1

3 11 1

20

( ) ( ) ( )

( )

k

k k

=

log logx

xdx

x x

x

x dx

e ex x1 130

1

31

20

.9520569031595942854... ( )31

4

1 12 5

2

k k k kk

3

= ( ) ( )k kk

2 12

.95217131705368432233... 2 2

2 2 0

4

logcos log(sin )

/

x x

dx

.952380952380952380 = 20

21

1

200

( )k

kk

... 1

4

1

22

1

2

1

2 2 1 20

, ,( )

( )

k

kk k

2 .95249244201255975651... ( )! !

ej kkj

112

11

.95257412682243321912... e

eek k

3

33

1

1

21

3

21

tanh ( ) J944

...

2

1

1 220e t

tdt

cos

... dxx

x

0

224

2cos2log

1

2

)4(11

180

11


... 551

2101

3

5

0

ek

kk ( )!

.954085357876006213145... ( )

!

k

kk

2

... Eik k

k

k

( ) log!

2 22

1

AS 5.1.10

... 1 4 5 ( ) ( )

... )3()2(23

)3(2

... 3

23 2 1

1

20

1

log log log

x

dx

... 2 100

2

I e x( ) sin dx

.95492965855137201461... 3

11

36 21

kk

GR 1.431

= cos

6 21

kk

GR 1.439.1

= 0

1 6/

2 .95555688550730281453... !

1)(2

2 k

k

k

k

k

... 7

128

3 3

42 1 13

1

( )( )k k

k

=k k k

kk

( )

( )

4 2

2 42

4 1

1

... 2931587

2 3

6

0

ek

kk

( )!

... 5

162

33

31

sin /k

kk

GR 1.443.5

.95706091455937067489... G

... log ( ) snns

12

.957657554360285763750... 1

22

1

2

1

2 11

cot tank kk

Berndt ch. 31

...

0

2

2

2

)1log(

2

1arcsinh

x

x

2980 .95798704172827474359... e8

...

22 2

2

1

21 1

1

81 1 ( ) ( ) ( ) ( )( ) ( )i i i i

5

81 1 21 1i

i i ( ) ( )( ) ( ) (3 )

1

= ( ) ( ) ( )

1 22

2

k

k

k k k

.95833608906520793715... ( )

( )!

1 1

21

k

k k

.95835813283300701621... cos cosh( )

( )!

1

2

1

2

1

40

k

k k

.95838045456309456205... 1

1024

5

8

3

82

1

82

8 82 2 2 3 2

( ) ( ) ( ) cot csc

= 11

4 1

1

4 13 31

( )

( )

( )

( )

k k

k k k

.95853147061909644370... 1

3

3

2 3

1 1

3 10

cos( )

( )

e k

k

k !

... 1 2

21

2

2 20

tanh( / )( )

/

/ /

e

e eek k

k

J944

2 .95867511918863889231...

1

43

4

... 21

2

1

2 1 41

1

402 2

1

sin( )

( )!

k

kk kk k

GR 1.431

= 0

1 2/

.95924696800225890378... 4 11

2 1

1

21

cik k

k

k

( )( )

( )!

... 1

2 1 2 11 2

0sinh( / )( / )

e

ee k

k

J942

.95974233961488293932...

03

3/2

233

2

x

dxx

.95985740794367609075... 2

12sin

1

2

1sin1cos22log

2

1cos)1(

2

1

1

k

kk

.9599364907945658556...

1

1)1()(k

kk

... k k

kk

( ( ) )

!

2

2

1

.96031092683032321590... 16

17

1

4

1

2 1 4

1 2

0

arcsinh

( ) ( !)

( )!

k

kk

k

k

.9603271579367892680... 4

4 21

2

1

1 4 2 11 40e

erfk k

k

kk

/

( )

( )! (

)

... 32 2

151

1

16 22

kk

... 2 22 2

0

2

48 8

2

4

2

4

log loglogsin cos

/

x x dx

... 2 22 1

2

1

log( )

H

k kk

k

... ( ) ( )

1

2 1

1

1

k

kk

k

... 2 21

2 2

1

8 2 10

arctan( )

( )

k

kk k

... 3 3 2 ( ) ( )

... 4 3

5

( )

... ( )

( )!

2 1

113

1

2

k

k

e

kk

k

k

... 44

5

1

2

1

4 2 1 10

log arctan( )

( )(

k

kk k k )

... 21

22

1 5

2

1

16 2 1

2

0

arcsinh

log( )

( )

k

kk k

k

k

... log 7 22 3

1

33 3

arccot

= 11

3

1

6 1

1

6 13 11

( )k

kk k k

... 2 5 ( )

... 5 1 1 1 11 5 2 5 3 5 4 5 1 ( ) ( ) ( ) ( )/ / /

/

= 11

52

kk

... 5

8

...

2 2 1

1

2

1 2

1 21log

( / )

( / )

=

3

21

.96400656328619110796... ( log )log( )

( )1 2

1

1

2

2 20

x

x xdx

... tanh22 2

2 2

e e

e e

3 .9640474536922083937... ( ) ( ) ( ) (i i i )i 1 1

.9641198407218830887... 26

4 23 3

4

1

2

2 1

4 31

log( ) ( )k

k k k

... 1

5 51

( )

( )

k

kk

...

( )

( )

( )10

5 51

k

kk

HW Thm. 300

.96558322350755543793... 1

2 20k

k

...

( )2

2 12 11

kk

k

... 2

41 3

5

12 sin

AS 4.3.46

... 6 5 3

108 16 20

log

( )

x

xdx

... erfk k

k k

k

3

2

2 1 3

2 2 1

2 1

2 1

( )

! ( )

... ( ) ( ) ( ) ( )2 3 4 5

... 1

4

1

23

1

2 1

2

21

, ,log

/

x

xdx

2

... 1

21 1 1 1

0

1

cosh sin cos sinh cos cosh x x dx

... 11

1

kk !!

...

1

0

1

0

1

0

1

02222

3

12log24

8dzdydxdw

zyxw

zyxwG

.96676638530855214796... 7

71

1

49 21

sin

kk

GR 1.431

= 0

1 7/

... ( )( ) ( )

11

212

2

k

k

kk k

.96761269589907614401... ( )4 3

2 14 31

kk

k

.9681192775644117... 5

.96854609799918165608... 1

64

1

8

5

8

1

4 11 1

20

( ) ( ) ( )

( )

k

k k

=

log logx

xdx

x x

xdx

1 140

1 2

41

= x x

x xdx

log3 1

1

3 .96874800690391485217... F Hk k

kk 21

4 .96888875288688326456... 2

2 2

( )

( )

k

kk

!

.96891242171064478414... cos( )

( )!

1

4

1

2 160

k

kk k

.96894614625936938048...

3

3032

31

2 1

( )( )

( )

k

k k AS 23.2.21

= Im ( ) ( ) ( )Li ii

Li i Li i3 32 3

= sin /k

kk

23

1

=

1

0

1

0

1

02221 zyx

dzdydx

... 2

2 11

k

k k k! ( )

Titchmarsh 14.32.3

.96936462317800478923...

2 2 2 121

2

csc

=( )

1

2

1

4 21

k

k k k

... 4

729 3

5

5

g J310

... 2

... 2 2

... G1 3/

=43867

798 9 B

...

2

0

2

0 sin2

sin

cos2

cos1

3

22

x

dxxdx

x

x

... k k kk

( ) ( )

1 12

... 15 5

165

11 5 0

1

51

( )( )

( )( , , )

k

k k AS 23.2.19

... k k kk

( ) ( )

1 12

... 276 150 22

4

1

logk Hk

kk

... 15 5

16

1

1 50

( ) ( )

( )

k

k k

... 61

6

1

6 2 10

sin( )

( )

k

kk k !

= 11

6 2 21

kk

... log7

2

= 11

3 6 1

1

3 6 13 1 31

( )

( )

( )

( )

k

k

k

kk k k

K ex. 108

1 .97318197252508458260... 1

216

2

3

1

62 2

2

20

( ) ( )

x dx

e ex x

= ( )( ) ( ) ( ) (/ / / /3

2

2

31 1 11 3

32 3 2 3

3

1 3 Li Li 1

= log log log2

30

1 2

2 11

2

311 1

x

xdx

x

x xdx

x x

xdx

.97336024835078271547...

1

3

1

2

2

3

2

32 3( ) /

2 .97356555791412288969... 4 2 3 3 ( ) ( )

.97402386878476711062... S k k

kk

2

1

2

2

( , )

( )!

.97407698418010668087... log( )1 e

... dxexx

x

e

0

22

))(1(

log

33

1

.9744447165890447142... 1

4

1

42

1

2

1

4 2 1 2 10

, ,( )

( )( )

k

kk k k

.97449535840443264512... 8

8 2 31

sin cos

kk

GR 1.439.1

.97499098879872209672... 1

22 1 2

1

3 40

log( )

/

k

k k

= dx

x x( ) cosh /1 420

GR 3.527.10

... 3

64

2

16

2

16 4

4 2 2 2

2 20

log log log x

x xdx

... sin Im ( )

2

2i

1 .97553189198547105414... 2 21 16

22

1

1

sin( )

( )!

k k

k k

... 4

1524 4

2

4 4

0

( )x

e ex x

... I HypPFQ0 12

13

2

3

2

1

4( ) { }, , ,

= e dxsin

0

1

x

... 11

360

3

24

1

22

1

42

2

21

( ) ( )( )

H

kk

k

18 MI 4, p. 15

.97645773647066825248... sin( )

( )!(

1

2

1

2 1 2 11

1

2 11

kk

k

kk k

)

...

2

2 211

1

21

1

2 1

cos( )kk

2

... J314 h2

1 .9773043502972961182... ( ) ( )

( ) ( ) ( )2 3

3

6

21

31

12

1

k

k

k

kk k

HW Thm. 290

...

I e dx ex x0

0

2

0

1( ) cos cos dx

2 .9775435572485657664...

12 24

2 22

2

2 2 2

log log

log

= log sin2 2

20

x x

xdx

4 .9777855916963031499... e I Ik

k

k

kkk

0 1 1 10

1 13

22 2

2

2( ) ( ) ( , , )

!

F

... 5535

2

22

3

0

k

k

kk

k

... 1

100

1

10

3

5

1

5 11 1

20

( ) ( ) ( )

( )

k

k k

=x x

x xdx

log3 2

1

1 .9781119906559451108... 2

2 2

06

log xdx

ex GR 4.355.1

.97829490163210534585... G1 4/ ... 40320 9 18 ( ) ( )( )

... Li2

3

4

.97917286680253558830... cos cosh( )

( )!/ /

1

2

1

2

1

4 23 4 3 40

k

kk k

.97929648814722060913... 2 21

2 2

1

2 10

sin( )

( )!

k

kk k 8

.97933950487701827107... 161

4

1

2 1 4 12

0

sin( )

( )! (

k

kk k k )

.97987958188123309882... loglog( )1 3

2

1

1

2

20

1

x

xdx GR 4.295.38

.98025814346854719171... 2 21

2 2

2 1

32 2 10

arcsinh

k

k k

k

kk

( )

( )

... 31

3

1

9 2 10

sin( )

( )

k

kk k !

= 11

9 2 21

kk

.98174770424681038702... 5

16 1

5 2

x

xdx

/

GR 3.226.2

.98185090468537672707... ( )

!

( )

( )!!/2

2

2

21

1 1

1 2

1

2k

k

k

kek

k k

k

k

.98187215105020335672... iLi Li

21 12

3 42

1 4 ( ) ( )/ /

= sin / k

kk

42

1

... 6

5

2

5 5

3

5

3

5 1 5 30

csc /

dx

x

... 1 2

21

2

2 24

0

tanh( )

e

e eek k

k

J944

... 19

27 3

1 3 3

0

e k

k kk

/

!

... 22

9

2 2

3

2

3

2 2 12

2

30

log e

x x

e

x

dx

x

x x

GR 3.438.2

... 2 6 ( )

... ( )

( )

12

2 1 30

k

k k

kk

... 1 3

12

1

6 2

3

21

12 2

26

2

coshsin

ikk

... 945 1

66 61

( )

( )k

kk

...

sin1

114 2

1

kk

... 1

8

1

23

1

2 2

2

20

1

, ,log x

x

=i

Lii

Lii

2 23 3

2

.983194483680076021738... 691

675675

1

1

6

6

pp prime

... ( ) ( ) ( ) ( ) ( )2 3 4 5 6

.98361025039925204067... 2 16

2 21

2 1

3 1

31

Gk k

k

k

log( )

( )

.98372133768990415044... 1

144

1

12

7

12

1

6 11 1

20

( ) ( ) ( )

( )

k

k k

= x x

x xdx

log3 3

1

10 .98385007371209431669... 2 4 2 2 3 23 3

1

2

32

( ) ( )( )

k k

k kk

1

= ( ) ( )k k kk

2

2

1

.98419916936149897912... 403

393660

6

6

J312

... 2

636 5

3 ( ) ( )

.9845422648017221585... 2

22

20

1

62 2

3

4

1 1

log

( ) ( ) log ( )x x

xdx

.98481748453515847115... ( )

14

1

k

kk k

.98517143100941603869... 2 2 1 21 4 / , from elliptic integrals

... log1

3

.9855342964496961782... 96 16

15 4

24 4

1

( )

( )k

kk

.9855510912974351041... 31

302406

31 6

32

16 1

61

( )( ) ( )k

k k J306

... 4 23

8 8

11

43

40

logsin logsin( )

( )( )

k

k k k

... 1

512

5

8

1

8 12 2

2

40

1

( ) ( ) log

x

xdx

.98621483608613337832... 21

2

1

2 1 4 2 10

sik k

k

kk

( )

( )! ( )

6 .98632012366331278404... 5 9

4

2

3

2 2

0

e k

k k

k

k

!( )

.9865428606939705039... 11

768 21

1

4 1

1

4 1

4

41

( )

( )

( )

( )

k k

k k k 4 J327, J344

.98659098626254229130... 1

97210 3 351 3

1

432

2

3

1

63 2

( ) ( ) ( )2

= ( )

( )

1

3 1 30

k

k k

12 .98673982624599908550... 28402

2187

1

2

1 10

1

( )k

kk

k

.98696044010893586188... 2

10

... 1856

243

380

81

4

3 4

4

1

logk Hk

kk

... sin 2

.98787218579881572198... 4

105

7 15

42015 15

44 4

csc csch

= ( )

1

1542

k

k k

... 2

156

1

1

4 3

0

1

( ) logx x

xdx

...

2

3 12018 3 1 2 1

log x

xdx

.98861592946536921938... 3 23 1 1

1

144 21

kk

GR 1.431

= cos

3 2 21

kk

= 0

1 12/

... sin sinh ( )( )

1 1 12

4 2

2 1

0

!

kk

k k GR 1.413.1

= 114 4

1

kk

... 1

1536

1

4

3

43 3 ( ) ( )

= i

Li i Li i2 4 4( ) ( )

=( )

( )

sin /

1

2 1

24

04

1

k

k kk

k

k

1 .98905357643767967767... 3 2

609 3 1

log x dx

x

.9890559953279725554... 2 2 2

202 12

2

log ( )x dx

e x

...

( )log

22

21

k

kk

... 0

1 2( )kk

kk

.98958488339991993644... cos cosh( )

( )!

1

2

1

2

1

4 40

k

kk k

.98961583701809171839... 41

4

1

2 1 160

sin( )

( )!

k

kk k

= 11

16 2 21

kk

.98986584618905381178... 41

4

2 1

64 2 10

arcsinh

k

k k

k

kk

( )

( )

1 .98999249660044545727... 1 3 23

2

1 9

22

1

1

cos sin( )

( )!

k k

k k

.99004001943815994979... 1

3456

5

12

7

122

1

12

7 4 3

4322 2 2

3

( ) ( ) ( ) ( )

= 11

6 1

1

6 13 31

( )

( )

( )

( )

k k

k k k

... 10

3

74

81 3 2

3

1

kk

kk

78 .99044152834577217825... 1212 1

1

10

0

e

k

k

k

k

( )

( )!

.99076316985337943267... k kk

( )

12

2

.99101186342305209843... 1

41 1( ) ( ) ( ) ( )i i i i

21 .99114857512855266924... 7

4 .9914442354842448121... 6

3 ( )

.99166639109270971002... HypPFQkk

, , , , ,( )!

1

5

2

5

3

5

4

5

1

3125

1

50

.99171985583844431043... 1

7 71

( )

( )

k

kk

6 .99193429822870080627... ( )kk

2

7

.99213995900836... 7

.99220085376959424562... 4

125

2 53

31

sin /k

kk

GR 1.443.5

.99223652952251116935... 56

94815 3

7

7

g J310

1 .992294767124987392926...

1

2

30,2,

1

)1(

)1(

kke

k

ee

ee

... )()(768 55

5

iLiiLii

... 63 7

647

1 1

71

( )( )

( )

k

k k AS 23.2.19

... 2

1 1

0

( ) sin(tan )

e xdx

x GR 3.881.2

.99297974679457358056... ( )3 2

2 13 21

kk

k

.99305152024997656497... 1

1024

5

8

1

82 2 ( ) ( )

... ii eLieLi 2

22

2

.9944020304296468447... 6 2 2 26 2

22 3

21

log log( )

k

kkk 1

.99452678821883983884... 3

318 3 h J314

1 .9947114020071633897... 5

2

.99532226501895273416... erfk k

k k

k

22 1 2

2 1

2 1

0

( )

!( )

1 .9954559575001380004... dx

x x0

...

2

20

( log ) log sin

x xdx

x GR 4.421.1

... ( ) ( )

( )

( )

( )

91 45 5

187501

1

5 1

1

5 1

4

4 41

k k

k k k

10 .99557428756427633462... 7

2

21 .99557428756427633462... 7

211

22

2

1

k

k

kk

k

... 1

786432

3

8

5

8

1

8

7

84 4 4 4 ( ) ( ) ( ) ( )

= 11

4 1

1

4 15 51

( )

( )

( )

( )

k k

k k k

... 2

2 2

( )kk

k

... arctan k

kk 21

.99592331507778367120...

sinhsin ( ) sin ( )/ /

81 1 1

13

1 4 3 48

2 kk

7 .99601165442664514565... ( )kk

2

8

1 .99601964118442342116... 142 3844

3

0

ek

k kk

!( )

.99608116021237162307... 5207

49601160

8

8

J312

... 32

3

238

81 3 2

4

1

kk

kk

.99610656865... J310 g8

.99615782807708806401... 5

15365

1

2 1

5

50

( )( )

( )

k

k k AS 23.2.21, J345

.99623300185264789923... 127

12096008

127 8

128

18 1

81

( )( ) ( )k

k k J306

.. 219

4096 2

5

= 11

4 1

1

4 15 51

( )

( )

( )

( )

k k

k k k

... tanhtan

i

i

e e

e e

... log

/

( )

( ' )

/3

4 3 4 1

22 20

1 2

K k k

k k

dk GR 6.153

8 .99646829575509185269... 2 2 9 31

33

0

log ( )sin

cos

x x d

x

x

... ( )( )

12

2

2

kk

k

k k

... e e e e

e e

k

ekk

( )

( ), ,

4 3 2

6

5

1

26 66 26 1

1

15 0

... 3

8

11

22

7 3

4

16 6 1

2 2 1

2 2

32

log( )

( )

k k

k kk

=k k

kk

2

2

1

2

( ( ) )

... ( )kk

2

9

... 9 J312

.9980501956570772372... 3236

55801305 3

9

9

g J310

... 23

1296 31

1

6 1

1

6 1

4

4 41

( )

( )

( )

( )

k k

k k k

.99809429754160533077...

( )( ) ( )

8255 9

256

1 1

91

k

k k AS 23.2.19

.99813603811037497213... 1

2 11

2

22

tanh( )

e

eek k J944

.9983343660981139742... 33 3

1

3

128 2

1

64

1

8 5

1

8 3

( ) ( )k kk

17 .99851436155475931539... 1016

81

512

27

4

3

1 2 3

41

log( )( )( )k k k Hk

kk

... 3 2 11

220

logx

dx

... 19

2 3 5 31

1

4 1

1

4 1

2 4

14 61

( )

( )

( )

( )

k k

k k k 6 J327

... 9 2

... HypPFQk

k

k

, , , , , ,( )

( )!

1

6

1

3

1

2

2

3

5

6

1

46656

1

60

... ( )( )

( )( ) ( )6

1

491520

1

4

3

4

1

2 15 5

60

k

k k AS 23.2.21

.998777285931944... J314 h4

33 .99884377071514942382... 5 1

8

4

2

4

0

e k

k k

k

k

!( )

... 48983

4591650240

10

10

... 73

684288010

511 10

512

110 1

101

( )( ) ( )k

k k J306

... 6 2

.99915501340010950975...

4

4 42

1 1

11

4sin sinh

kk

4

... e e e

e

k

k

e e

k

2 1 2

1

1

2 1

/ sinh

( )!

... 7

11

1 .99935982878411178897... 3 2

120

7 4 3

216 1

log x dx

x

... 3 2

... 61

1843207

1

2 1

7

70

( )( )

( )

k

k k AS 23.2.21

... 2307 2

13107201

1

4 1

1

4 1

7

7 71

( )

( )

( )

( )

k k

k k k

... 11

1944 3

5

5

h J314

... 2305

933121

1

6 1

1

6 1

5

5 51

( )

( )

( )

( )

k k

k k k

.9997539139218932560...

sinhcosh sin ( ) sin ( )/ /

12

3

21 15

2 1 6 5 6

= 1112

2

kk

.99980158731305804717... ( )

( )!

1

70

k

k k

... 24611

165150720 21

1

4 1

1

4 1

8

81

( )

( )

( )

( )

k k

k k k 8 J327

.99984999024682965634... ( ) ( ) ( )81

330301440

1

4

3

47 7

=( )

( )

1

2 1 80

k

k k

... 1681

933120 31

1

6 1

1

6 1

6

61

( )

( )

( )

( )

k k

k k k 6 J327

... 6385

412876809

1

2 1

9

90

( )( )

( )

k

k k AS 23.2.21

... ( )

( )!

1

80

k

k k

... 301

524880 3

7

7

h J314

... 23337

1007769601

1

6 1

1

6 1

7

7 71

( )

( )

( )

( )

k k

k k k

... ( )

( )!

1

90

k

k k

...

188

8

)16(

)1(

)16(

)1(1

31410877440

25743

k

kk

kk

J329

.99999972442680776055... ( )

( )

1

100

k

k k !

A Catalog of the Real Numbers - Carnegie Mellon...

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